Class 11 • Physics

Gravitation

Chapter-7

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MediumAiims2019

Find gravitational field at a distance of $2000 \mathrm{~km} from the centre of earth. (Given R_{\text {earth }}=6400 \mathrm{~km}, r=2000 \mathrm{~km} \text {, } M_{\text {earth }}=6 \times 10^{24} \mathrm{~kg} \text { ) }

Options:

A) 1.53 \mathrm{~m} / \mathrm{s}^2

B) 7.12 \mathrm{~m} / \mathrm{s}^2

C) 3.06 \mathrm{~m} / \mathrm{s}^2

D) 1.8 \mathrm{~m} / \mathrm{s}^2

MediumAiims2018

Two satellites $A and B revolve round the same planet in coplanar circular orbits lying in the same plane. Their periods of revolutions are 1 \mathrm{~h} and 8 \mathrm{~h}, respectively. The radius of the orbit of A is 10^4 \mathrm{~km}. The speed of B is relative to A. When they are closed in \mathrm{km} / \mathrm{h}$ is

Options:

A) 3 \pi \times 10^4

B) zero

C) 2 \pi \times 10^4

D) \pi \times 10^4

MediumAiims2018

A planet is revolving around the sun in a circular orbit with a radius $r. The time period is T. If the force between the planet and star is proportional to r^{-3 / 2}$, then the square of time period is proportional to

Options:

A) r^{3 / 2}

B) r^2

C) r

D) r^{5 / 2}

MediumAiims2018

The weight of a body on the surface of the earth is 63 N. What is the gravitational force on it due to the earth at a height equal to half the radius of the earth?

Options:

A) 35 N

B) 28 N

C) 18 N

D) 40 N

MediumAiims2017

A space ship is launched into a circular orbit close to earth’s surface. What additional velocity has now to be imparted to the spaceship in the orbit to overcome the gravitational pull? (Radius of earth = 6400 km, g = 9.8 m/s$^2$)

Options:

A) 3.28 km/s

B) 12 km/s

C) 10 km/s

D) 40 km/s

MediumAiims2017

What is the maximum height attained by a body projected with a velocity equal to one-third of the escape velocity from the surface of the earth? (Radius of the earth $=R$ )

Options:

A) R / 2

B) R / 3

C) R / 5

D) R / 8

MediumAiims2017

Two satellites $S_1 and S_2 are revolving round a planet in coplanar circular orbits of radii r_1 and r_2 in the same direction, respectively. Their respective periods of revolution are 1 \mathrm{~h} and 8 \mathrm{~h}. The radius of orbit of satellite S_1 is equal to 10^4 \mathrm{~km}. What will be their relative speed (in \mathrm{km} / \mathrm{h}$) when they are closest?

Options:

A) \pi / 2 \times 10^4

B) \pi \times 10^4

C) 2 \pi \times 10^4

D) 4 \pi \times 10^4

MediumBITSAT2024

A body which is initially at rest at a height R above the surface of the Earth of radius R , falls freely towards the Earth, then its velocity on reaching the surface of the Earth is

Options:

A) \sqrt{(2 g R)}

B) \sqrt{(g R)}

C) \sqrt{\frac{3}{2} g R}

D) \sqrt{(4 g R)}

MediumBITSAT2023

A man of mass m starts falling towards a planet of mass $M and radius R. As he reaches near to the surface, he realizes that will pass through a small role in the planet. As he enters the role, we sees that the planet is really made of two pieces a spherical shell of negligible thickness of mass 3 M / 4 and a point mass M / 4$ at the centre. Change in the force of gravity experienced by the man is

Options:

A) \frac{3}{4} \frac{G M m}{R^2}

B) 0

C) \frac{1}{3} \frac{G M m}{R^2}

D) \frac{4}{3} \frac{G M m}{R^2}

MediumBITSAT2022

What will be the acceleration due to gravity at a depth d, where g is acceleration due to gravity on the surface of earth?

Options:

A) {g \over {{{\left[ {1 + {d \over R}} \right]}^2}}}

B) g\left[ {1 - {{2d} \over R}} \right]

C) {g \over {{{\left[ {1 - {d \over R}} \right]}^2}}}

D) g\left[ {1 - {d \over R}} \right]

MediumBITSAT2021

From a solid sphere of mass M and radius R, a spherical portion of radius ${R \over 2} is removed as shown in the figure. Taking gravitational potential V = 0 at r = \infty$, the potential at the centre of the cavity thus formed is

Options:

A) - {{GM} \over {2R}}

B) - {{GM} \over {R}}

C) - {{2GM} \over {3R}}

D) - {{2GM} \over {R}}

MediumBITSAT2021

If the earth stops rotating about its axis, then what will be the change in the value of g at a place in the equatorial plane? (Radius of earth = 6400 km)

Options:

A) 3.7 cm/s2

B) 9.8 m/s2

C) 0

D) 3.4 cm/s2

MediumBITSAT2020

Two spheres of the same material and same radii r are touching each other. The gravitational force between the spheres is proportional to

Options:

A) {1 \over {{r^2}}}

B) r2

C) {1 \over {{r^4}}}

D) r4

MediumBITSAT2020

A satellite is moving around the earth with speed v in a circular orbit of radius r. If the orbit radius is decreased by 2%, its speed will increase by

Options:

A) 1%

B) 2%

C) 1.5%

D) 1.414%

MediumCOMEDK2025

The unit of universal gravitational constant is :

Options:

A) \mathrm{N} \mathrm{m}^2 \mathrm{~kg}^2

B) \mathrm{N} \mathrm{m}^{-2} \mathrm{~kg}^{-2}

C) \mathrm{N} \mathrm{m}^{-2} \mathrm{~kg}^2

D) \mathrm{N} \mathrm{m}^2 \mathrm{~kg}^{-2}

MediumCOMEDK2025

If the distance between the Sun and Earth is doubled, then the duration of the year on earth will be : [Given actual duration of the year =\mathbf{T} ]

Options:

A) 2 \sqrt{2} T

B) \frac{T}{2}

C) \sqrt{\mathbf{2}} \boldsymbol{T}

D) \frac{T}{\sqrt{2}}

MediumCOMEDK2025

The value of universal gravitational constant was first determined by

Options:

A) Newton

B) Cavendish

C) Einstein

D) Galileo

MediumCOMEDK2025

Two spherical planets P and Q have the same uniform density \rho, and masses Mp and \mathrm{MQ}_{\mathrm{Q}} and surface areas A and 4 A respectively. Another spherical planet R also has the same uniform density \rho, and its mass is \mathrm{Mp}+\mathrm{MQ}. The escape velocities from these planets is

Options:

A) V_R>V_Q>V_P

B) \mathrm{V}_{\mathrm{R}}<\mathrm{V}_{\mathrm{Q}}<\mathrm{V}_{\mathrm{P}}

C) V_R=V_Q>V_P

D) V_R>V_Q=V_P

MediumCOMEDK2025

If the radius of earth were to shrink by two percent, its mass remaining the same, the acceleration due to gravity on the earth's surface would

Options:

A) Decrease by 4 \%

B) Increase by 1 \%

C) Increase by 4 \%

D) Increase by 2 \%

MediumCOMEDK2025

The radius of earth is R and acceleration due to gravity on its surface is g. The height at which the acceleration due to gravity becomes \frac{g}{8} is:

Options:

A) 2 R

B) (2 \sqrt{2}-1) R

C) \sqrt{2} R

D) 2 \sqrt{2} R

MediumCOMEDK2025

A planet is 121 times heavier than moon and has a diameter 9 times that of moon. If the escape velocity on the planet is v, then the escape velocity on the moon will be:

Options:

A) \frac{11 v}{3}

B) \frac{33 v}{8}

C) \frac{8 v}{33}

D) \frac{3 v}{11}

MediumCOMEDK2024

The acceleration due to gravity at pole and equator can be related as

Options:

A) g_e=g_p < g

B) g_e=g_p=g

C) g_e < g_p

D) g_e > g_p

MediumCOMEDK2024

A satellite is revolving around the earth in a circular orbit with kinetic energy of $1.69 \times 10^{10} \mathrm{~J}$. The additional kinetic energy required for just escaping into the outer space is

Options:

A) 3.38 \times 10^{10} \mathrm{~J}

B) 1.69 \times 10^{10} \mathrm{~J}

C) 0.89 \times 10^{10} \mathrm{~J}

D) 1.35 \times 10^{10} \mathrm{~J}

MediumCOMEDK2024

A planet has double the mass of the earth and double the radius. The gravitational potential at the surface of the Earth is $\mathrm{V} and the magnitude of the gravitational field strength is \mathrm{g}. The gravitational potential and gravitational field strength on the surface of the planet are Potential Field A V \frac{g}{4} B 2V \frac{g}{2} C V \frac{g}{2} D 2V \frac{g}{4}

Options:

A) C

B) D

C) A

D) B

MediumCOMEDK2024

Energy required for moving a body of mass $\mathrm{m}$ from a circular orbit of radius 3R to a higher orbit of radius 4R around the earth is.

Options:

A) \frac{\mathrm{GMm}}{\mathrm{R}}

B) \frac{\mathrm{GMm}}{24 \mathrm{R}}

C) \frac{\mathrm{GMm}}{4 \mathrm{R}}

D) \frac{\mathrm{GMm}}{12 \mathrm{R}}

MediumCOMEDK2024

If $\mathrm{A} is the areal velocity of a planet of mass \mathrm{M}$, then its angular momentum is

Options:

A) \frac{\mathrm{MA}}{2}

B) MA

C) 2 \mathrm{MA}

D) \frac{\mathrm{MA}}{3}

MediumCOMEDK2024

If the earth has a mass nine times and radius four times that of planet X, the ratio of the maximum speed required by a rocket to pull out of the gravitational force of planet $\mathrm{X}$ to that of the earth is

Options:

A) \frac{2}{3}

B) \frac{9}{4}

C) \frac{3}{2}

D) \frac{4}{9}

MediumCOMEDK2023

Starting from the centre of the earth having radius $R, the variation of g$ (acceleration due to gravity) is shown by

Options:

MediumCOMEDK2023

If escape velocity on earth surface is $11.1 \mathrm{~kmh}^{-1}, then find the escape velocity on moon surface. If mass of moon is \frac{1}{81} times of mass of earth and radius of moon is \frac{1}{4}$ times radius of earth.

Options:

A) 2.46 \mathrm{~kmh}^{-1}

B) 3.46 \mathrm{~kmh}^{-1}

C) 4.4 \mathrm{~kmh}^{-1}

D) None of these

MediumCOMEDK2023

The height vertically above the earth's surface at which the acceleration due to gravity becomes $1 \%$ of its value at the surface is

Options:

A) 8R

B) 9R

C) 10R

D) 20R

MediumCOMEDK2023

An uniform sphere of mass $M and radius R exerts a force of F on a small mass m placed at a distance of 3R from the centre of the sphere. A spherical portion of diameter R is cut from the sphere as shown in the fig. The force of attraction between the remaining part of the disc and the mass \mathrm{m}$ is

Options:

A) 7F/12

B) F/3

C) 41F/50

D) 7F/9

MediumCOMEDK2023

The acceleration due to gravity at a height of $7 \mathrm{~km}$ above the earth is the same as at a depth d below the surface of the earth. Then d is

Options:

A) 7 km

B) 2 km

C) 3.5 km

D) 14 km

MediumCOMEDK2022

If the earth were to spin faster, acceleration due to gravity at the poles

Options:

A) increases

B) decreases

C) remains the same

D) depends on how fast it spins

MediumCOMEDK2022

The height at which the acceleration due to gravity becomes $\frac{g}{16}$ (where, g = acceleration due to gravity on the surface of the earth) in terms of R is, if R is the radius of earth.

Options:

A) 2R

B) 3R

C) \sqrt2R

D) \sqrt3R

MediumCOMEDK2022

The escape velocity of a projectile on the earth's surface is 11.2 km/s. A body is projected out with thrice this speed. The speed of the body far away from the earth will be

Options:

A) 22.4 km/s

B) 31.7 km/s

C) 33.6 km/s

D) None of these

MediumCOMEDK2021

Kepler's second law of planetary motion corresponds to

Options:

A) conservation of energy

B) conservation of angular momentum

C) conservation of linear momentum

D) conservation of mass

MediumCOMEDK2021

A constant potential energy of a satellite is given as $\mathrm{PE}=r(\mathrm{KE}) whee, PE = potential energy and KE = kinetic energy. The value of r$ will be

Options:

A) -1

B) -2

C) \frac{-1}{2}

D) \frac{-3}{2}

MediumCOMEDK2020

A satellite can be in a geostationary orbit around a planet if it is at a distance R from the centre of the planet. If the planet starts rotating about its axis with double the angular velocity, then to make the satellite geostationary, its orbital radius should be

Options:

A) 2R

B) \frac{R}{2}

C) \frac{R}{2^{1/3}}

D) \frac{R}{4^{1/3}}

MediumCOMEDK2020

Two spherical bodies of masses M and 5M and radii R and 2R are released in free space with initial separation between their centres equal to 12 R. If they attract each other due to gravitational force only, then the distance covered by the smaller body before collision is

Options:

A) 2.5 R

B) 4.5 R

C) 7.5 R

D) 1.5 R

HardJee Advance2025

Consider a star of mass m2 kg revolving in a circular orbit around another star of mass m1 kg with m1 \gg m2. The heavier star slowly acquires mass from the lighter star at a constant rate of \gamma kg/s. In this transfer process, there is no other loss of mass. If the separation between the centers of the stars is r, then its relative rate of change \frac{1}{r}\frac{dr}{dt} (in s−1) is given by:

Options:

A) -\frac{3\gamma}{2m_{2}}

B) -\frac{2\gamma}{m_{2}}

C) -\frac{2\gamma}{m_{1}}

D) -\frac{3\gamma}{2m_{1}}

HardJee Advance2024

A particle of mass m is under the influence of the gravitational field of a body of mass M(\gg m). The particle is moving in a circular orbit of radius r_0 with time period T_0 around the mass M. Then, the particle is subjected to an additional central force, corresponding to the potential energy V_{\mathrm{c}}(r)=m \alpha / r^3, where \alpha is a positive constant of suitable dimensions and r is the distance from the center of the orbit. If the particle moves in the same circular orbit of radius r_0 in the combined gravitational potential due to M and V_{\mathrm{c}}(r), but with a new time period T_1, then \left(T_1^2-T_0^2\right) / T_1^2 is given by [G is the gravitational constant.]

Options:

A) \frac{3 \alpha}{G M r_0^2}

B) \frac{\alpha}{2 G M r_0^2}

C) \frac{\alpha}{G M r_0^2}

D) \frac{2 \alpha}{G M r_0^2}

MediumJee Advance2023

Two satellites \mathrm{P} and \mathrm{Q} are moving in different circular orbits around the Earth (radius R ). The heights of \mathrm{P} and \mathrm{Q} from the Earth surface are h_{\mathrm{P}} and h_{\mathrm{Q}}, respectively, where h_{\mathrm{P}}=R / 3. The accelerations of \mathrm{P} and \mathrm{Q} due to Earth's gravity are g_{\mathrm{P}} and g_{\mathrm{Q}}, respectively. If g_{\mathrm{P}} / g_{\mathrm{Q}}=36 / 25, what is the value of h_{\mathrm{Q}} ?

Options:

A) \frac{3 R}{5}

B) \frac{R}{6}

C) \frac{6 R}{5}

D) \frac{5 R}{5}

MediumJee Advance2019

Consider a spherical gaseous cloud of mass density $\rho (r) in free space where r is the radial distance from its center. The gaseous cloud is made of particles of equal mass m moving in circular orbits about the common center with the same kinetic energy K. The force acting on the particles is their mutual gravitational force. If \rho (r) is constant in time, the particle number density n(r) = \rho $(r)/m is [G is universal gravitational constant]

Options:

A) {K \over {6\pi {r^2}{m^2}G}}

B) {K \over {\pi {r^2}{m^2}G}}

C) {3K \over {\pi {r^2}{m^2}G}}

D) {K \over {2\pi {r^2}{m^2}G}}

MediumJee Advance2018

A planet of mass $M, has two natural satellites with masses {m_1} and {m_2}. The radii of their circular orbits are {R_1} and {R_2} respectively, Ignore the gravitational force between the satellites. Define {v_1},{L_1},{K_1} and {T_1} to be , respectively, the orbital speed, angular momentum, kinetic energy and time period of revolution of satellite 1; and {v_2},{L_2},{K_2}, and {T_2} to be the corresponding quantities of satellite 2. Given {m_1}/{m_2} = 2 and {R_1}/{R_2} = 1/4, match the ratios in List-{\rm I} to the numbers in List-{\rm II}.$ LIST - I LIST - II P. v1/v2 1. 1/8 Q. L1/L2 2. 1 R. K1/K2 3. 2 S. T1/T2 4. 8

Options:

A) P \to 4;Q \to 2;R \to 1;S \to 3

B) P \to 3;Q \to 2;R \to 4;S \to 1

C) P \to 2;Q \to 3;R \to 1;S \to 4

D) P \to 2;Q \to 3;R \to 4;S \to 1

MediumJee Advance2017

A rocket is launched normal to the surface of the Earth, away from the sun, along the line joining the Sun and the Earth. The Sun is $3 \times 10{}^5 times heavier than the earth and is at a distance 2.5 \times {10^4} times larger than the radius of the Earth. The escape velocity from Earth's gravitational field is {V_c} = 11.2km\,{s^{ - 1}}.. The minimum initial velocity \left( {{v_s}} \right)$ required for the rocket to be able to leave the sun-earth system is closest to (Ignore the the rotation and revoluation of the earth and the presence of any other planet)

Options:

A) {v_s} = 22\,km\,{s^{ - 1}}

B) {v_s} = 42\,km\,{s^{ - 1}}

C) {v_s} = 62km\,{s^{ - 1}}

D) {v_s} = 72km{s^{ - 1}}

MediumJee Advance2014

A planet of radius R = ${1 \over {10}} \times (radius of Earth) has the same mass density as Earth. Scientists dig a well of depth {R \over 5} on it and lower a wire of the same length and of linear mass density 10-3 kg m-1 into it. If the wire is not touching anywhere, the force applied at the top of the wire by a person holding it in place is (take the radius of Earth = 6 \times $ 106 m and the acceleration due to gravity of Earth is 10 ms -2)

Options:

A) 96 N

B) 108 N

C) 120 N

D) 150 N

MediumJee Advance2011

A satellite is moving with a constant speed ‘V’ in a circular orbit about the earth. An object of mass ‘m’ is ejected from the satellite such that it just escapes from the gravitational pull of the earth. At the time of its ejection, the kinetic energy of the object is

Options:

A) {1 \over 2}m{V^2}

B) m{V^2}

C) {3 \over 2}m{V^2}

D) 2m{V^2}

MediumJee Advance2010

A thin uniform annular disc (see figure) of mass M has outer radius 4R and inner radius 3R. The work required to take a unit mass from point P on its axis to infinity is

Options:

A) {{2GM} \over {7R}}(4\sqrt 2 - 5)

B) - {{2GM} \over {7R}}(4\sqrt 2 - 5)

C) {{GM} \over {4R}}

D) {{2GM} \over {5R}}(\sqrt 2 - 1)

HardJee Advance2009

Column II shows five systems in which two objects are labelled as X and Y. Also in each case a point P is shown. Column I gives some statements about X and/or Y. Match these statements to the appropriate system(s) from Column II: Column I Column II (A) The force exerted by X on Y has a magnitude $Mg. (P) Block Y of mass M left on a fixed inclined plane X, slides on it with a constant velocity. (B) The gravitational potential energy of X is continuously increasing. (Q) Two rings magnets Y and Z, each of mass M, are kept in frictionless vertical plastic stand so that they repel each other. Y rests on the base X and Z hangs in air in equilibrium. P is the topmost point of the stand on the common axis of the two rings. The whole system is in a lift that is going up with a constant velocity. (C) Mechanical energy of the system X + Y is continuously decreasing. (R) A pulley Y of mass m_0$ is fixed to a table through a clamp X. A block of mass M hangs from a string that goes over the pulley and is fixed at point P of the table. The whole system is kept in a lift that is going down with a constant velocity. (D) The torque of the weight of Y about point is zero. (S) A sphere Y of mass M is put in a non-viscous liquid X kept in a container at rest. The sphere is released and it moves down in the liquid. (T) A sphere Y of mass M is falling with its terminal velocity in a viscous liquid X kept in a container.

Options:

A) \mathrm{(A)\to (T),(S);(B)\to (Q),(T);(C)\to(P),(R),(T);(D)\to(Q)}

B) \mathrm{(A)\to (T),(P);(B)\to (Q),(S),(T);(C)\to(P),(R),(T);(D)\to(Q)}

C) \mathrm{(A)\to (T),(Q);(B)\to (Q),(S),(T);(C)\to(P),(R),(T);(D)\to(S)}

D) \mathrm{(A)\to (P);(B)\to (S),(T);(C)\to(P),(R),(T);(D)\to(T)}

MediumJee Advance2008

A spherically symmetric gravitational system of particles has a mass density $\rho = \left\{ {\matrix{ {{\rho _0}} & {for} & {r \le R} \cr 0 & {for} & {r > R} \cr } } \right. Where \rho_0 is a constant. A test mass can undergo circular motion under the influence of the gravitational field of particles. Its speed V as a function of distance r(0 < r < \infty)$ from the centre of the system is represented by

Options:

EasyJee Advance2008

STATEMENT - 1 An astronaut in an orbiting space station above the Earth experiences weightlessness. and STATEMENT - 2 An object moving around the Earth under the influence of Earth's gravitational force is in a state of 'free-fall'.

Options:

A) Statement - 1 is True, Statement - 2 is True; Statement - 2 is a correct explanation for Statement - 1

B) Statement - 1 is True, Statement - 2 is True; Statement - 2 is NOT a correct explanation for Statement - 1

C) Statement - 1 is True, Statement - 2 is False

D) Statement - 1 is False, Statement - 2 is True

MediumJee Advance2007

Some physical quantities are given in Column I and some possible SI units in which these quantities may be expressed are given in Column II. Match the physical quantities in Column I with the units in Column II and indicate your answer by darkening appropriate bubbles in the 4 $\times 4 matrix given in the ORS. Column I Column II (A) GM_eM_sG - universal gravitational constant,M_e - mass of the earth,M_s - mass of the Sun (P) (volt)(coulomb)(metre) (B) {{3RT} \over M}R - universal gas constant,T - absolute temperature,M - molar mass (Q) (kilogram)(metre)^3(second)^{-2} (C) {{{F^2}} \over {{q^2}{B^2}}}F - force,q - charge,B - magnetic field (R) (metre)^2(second)^{-2} (D) {{G{M_e}} \over {{R_e}}}G - universal gravitational constant,M_e - mass of the earthR_e - radius of the earth (S) (farad)(volt)^2(kg)^{-1}

Options:

A) (A)→(P), (Q); (B)→(R), (S); (C)→(R), (S); (D)→(R), (S)

B) (A)→(P); (B)→(R), (S); (C)→(R), (S); (D)→(R)

C) (A)→(P), (Q); (B)→(S); (C)→(R), (S); (D)→(S)

D) (A)→(Q); (B)→(R), (S); (C)→(S); (D)→(R), (S)

MediumJee Advance2006

A system of binary stars of masses m_{\mathrm{A}} and m_{\mathrm{B}} are moving in circular orbits of radii r_{\mathrm{A}} and r_R, respectively. If \mathrm{T}_A and \mathrm{T}_B are the time periods of masses m_A and m_B respectively, then

Options:

A) \frac{\mathrm{T}_{\mathrm{A}}}{\mathrm{T}_{\mathrm{B}}}=\left(\frac{r_{\mathrm{A}}}{r_{\mathrm{B}}}\right)^{\frac{3}{2}}

B) \mathrm{T}_{\mathrm{A}}>\mathrm{T}_{\mathrm{B}} if \left(r_{\mathrm{A}}>r_{\mathrm{B}}\right)

C) \mathrm{T}_{\mathrm{A}}>\mathrm{T}_{\mathrm{B}} if \left(m_{\mathrm{A}}>m_{\mathrm{B}}\right)

D) \mathrm{T}_{\mathrm{A}}=\mathrm{T}_{\mathrm{B}}

HardJee Advance2025

A geostationary satellite above the equator is orbiting around the earth at a fixed distance r_1 from the center of the earth. A second satellite is orbiting in the equatorial plane in the opposite direction to the earth's rotation, at a distance r_2 from the center of the earth, such that r_1=1.21 r_2. The time period of the second satellite as measured from the geostationary satellite is \frac{24}{p} hours. The value of p is _________.

Options:

HardJee Advance2022

Two spherical stars A and B have densities \rho_{A} and \rho_{B}, respectively. A and B have the same radius, and their masses M_{A} and M_{B} are related by M_{B}=2 M_{A}. Due to an interaction process, star A loses some of its mass, so that its radius is halved, while its spherical shape is retained, and its density remains \rho_{A}. The entire mass lost by A is deposited as a thick spherical shell on B with the density of the shell being \rho_{A}. If v_{A} and v_{B} are the escape velocities from A and B after the interaction process, the ratio \frac{v_{B}}{v_{A}}=\sqrt{\frac{10 n}{15^{1 / 3}}}. The value of n is __________ .

Options:

MediumJee Advance2021

The distance between two stars of masses 3MS and 6MS is 9R. Here R is the mean distance between the centers of the Earth and the Sun, and MS is the mass of the Sun. The two stars orbit around their common center of mass in circular orbits with period nT, where T is the period of Earth's revolution around the Sun. The value of n is __________.

Options:

MediumJee Advance2015

A large spherical mass M is fixed at one position and two identical masses m are kept on a line passing through the centre of M (see figure). The point masses are connected by a rigid massless rod of length l and this assembly is free to move along the line connecting them.All three masses interact only through their mutual gravitational interaction. When the point mass nearer to M is at a distance r = 3l from M the tension in the rod is zero for m = $k\left( {{M \over {288}}} \right)$. The value of k is

Options:

MediumJee Advance2015

A bullet is fired vertically upwards with velocity v from the surface of a spherical planet. When it reaches its maximum height, its acceleration due to the planet’s gravity is ${\left( {{1 \over 4}} \right)^{th}} of its value at the surface of the planet. If the escape velocity from the planet is {v_{esc}} = v\sqrt N $, then the value of N is (ignore energy loss due to atmosphere)

Options:

MediumJee Advance2010

A binary star consists of two stars A (mass 2.2Ms) and B (mass 11Ms), where Ms is the mass of the sun. They are separated by distance d and are rotating about their centre of mass, which is stationary. The ratio of the total angular momentum of the binary star to the angular momentum of star B about the centre of mass is

Options:

MediumJee Advance2010

Gravitational acceleration on the surface of a planet is ${{\sqrt 6 } \over {11}}g, where g is the gravitational acceleration on the surface of the earth. The average mass density of the planet is {2 \over 3}$ times that of the earth. If the escape speed on the surface of the earth is taken to be 11 kms-1, the escape speed on the surface of the planet in kms-1 will be

Options:

MediumJee Advance2013

Two bodies, each of mass M, are kept fixed with a separation $2L$. A particle of mass m is projected from the midpoint of the line joining their centres, perpendicular to the line. The gravitational constant is G. The correct statement(s) is (are)

Options:

A) The minimum initial velocity of the mass m to escape the gravitational field of the two bodies is $4\sqrt {{{GM} \over L}}

B) The minimum initial velocity of the mass m to escape the gravitational field of the two bodies is $2\sqrt {{{GM} \over L}}

C) The minimum initial velocity of the mass m to escape the gravitational field of the two bodies is $\sqrt {{{2GM} \over L}}

D) The energy of the mass m remains constant.

MediumJee Advance2012

Two spherical planets P and Q have the same uniform density r, masses MP and MQ and surface areas A and 4A respectively. A spherical planet R also has uniform density r and its mass is (MP + MQ). The escape velocities from the planets P, Q and R are VP, VQ and VR, respectively. Then

Options:

A) VQ > VR > VP

B) VR > VQ > VP

C) {{{V_R}} \over {{V_P}}} = 3

D) {{{V_P}} \over {{V_Q}}} = {1 \over 2}

MediumJEE Mains2026

Three masses 200 \mathrm{~kg}, 300 \mathrm{~kg} and 400 kg are placed at the vertices of an equilateral triangle with sides 20 m . They are rearranged on the vertices of a bigger triangle of side 25 m and with the same centre. The work done in this process \_\_\_\_ J. (Gravitational constant \mathrm{G}=6.7 \times 10^{-11} \mathrm{~N} \mathrm{~m}^2 / \mathrm{kg}^2 )

Options:

A) 4.77 \times 10^{-7}

B) 1.74 \times 10^{-7}

C) 9.86 \times 10^{-6}

D) 2.85 \times 10^{-7}

MediumJEE Mains2026

Given below are two statements : Statement I : A satellite is moving around earth in the orbit very close to the earth surface. The time period of revolution of satellite depends upon the density of earth. Statement II : The time period of revolution of the satellite is T=2 \pi \sqrt{\frac{R_e}{g}} (for satellite very close to the earth surface), where R_{\mathrm{e}} radius of earth and g acceleration due to gravity. In the light of the above statements, choose the correct answer from the options given below :

Options:

A) Statement I is true but Statement II is false

B) Statement I is false but Statement II is true

C) Both Statement I and Statement II are true

D) Both Statement I and Statement II are false

EasyJEE Mains2026

The escape velocity from a spherical planet A is 10 \mathrm{~km} / \mathrm{s}. The escape velocity from another planet B whose density and radius are 10 \% of those of planet A, is \_\_\_\_ \mathrm{m} / \mathrm{s}.

Options:

A) 1000

B) 200 \sqrt{5}

C) 1000 \sqrt{2}

D) 100 \sqrt{10}

MediumJEE Mains2026

Net gravitational force at the center of a square is found to be F_1 when four particles having mass M, 2 M, 3 M and 4 M are placed at the four corners of the square as shown in figure and it is F_2 when the positions of 3 M and 4 M are interchanged. The ratio \frac{F_1}{F_2} is \frac{\alpha}{\sqrt{5}}. The value of \alpha is \_\_\_\_ .

Options:

A) 2

B) 2 \sqrt{5}

C) 1

D) 3

MediumJEE Mains2026

Initially a satellite of 100 kg is in a circular orbit of radius 1.5 \mathrm{R}_{\mathrm{E}}. This satellite can be moved to a circular orbit of radius 3 R_E by supplying \alpha \times 10^6 \mathrm{~J} of energy The value of \alpha is \_\_\_\_ . (Take Radius of Earth R_E=6 \times 10^6 \mathrm{~m} and \mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2 )

Options:

A) 500

B) 1000

C) 100

D) 150

EasyJEE Mains2025

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).Assertion (A): The radius vector from the Sun to a planet sweeps out equal areas in equal intervals of time and thus areal velocity of planet is constant.Reason (R): For a central force field the angular momentum is a constant.In the light of the above statements, choose the most appropriate answer from the options given below:

Options:

A) (A) is not correct but (R) is correct

B) Both (A) and (R) are correct but (R) is not the correct explanation of (A)

C) Both (A) and (R) are correct and (R) is the correct explanation of (A)

D) (A) is correct but (R) is not correct

MediumJEE Mains2025

An object is kept at rest at a distance of 3 R above the earth's surface where R is earth's radius. The minimum speed with which it must be projected so that it does not return to earth is : (Assume \mathrm{M}= mass of earth, \mathrm{G}= Universal gravitational constant)

Options:

A) \sqrt{\frac{3 \mathrm{GM}}{\mathrm{R}}}

B) \sqrt{\frac{2 \mathrm{GM}}{\mathrm{R}}}

C) \sqrt{\frac{\mathrm{GM}}{2 \mathrm{R}}}

D) \sqrt{\frac{\mathrm{GM}}{\mathrm{R}}}

MediumJEE Mains2025

Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason \mathbf{R} Assertion A : The kinetic energy needed to project a body of mass m from earth surface to infinity is \frac{1}{2} \mathrm{mgR}, where R is the radius of earth. Reason R : The maximum potential energy of a body is zero when it is projected to infinity from earth surface. In the light of the above statements, choose the correct answer from the options given below

Options:

A) \mathbf{A} is false but \mathbf{R} is true

B) Both \mathbf{A} and \mathbf{R} are true and \mathbf{R} is the correct explanation of \mathbf{A}

C) \mathbf{A} is true but \mathbf{R} is false

D) Both \mathbf{A} and \mathbf{R} are true but \mathbf{R} is NOT the correct explanation of \mathbf{A}

EasyJEE Mains2025

\text { Match the LIST-I with LIST-II } List - I List - II A. \text { Gravitational constant } I. \left[\mathrm{LT}^{-2}\right] B. \text { Gravitational potential energy } II. \left[\mathrm{L}^2 \mathrm{~T}^{-2}\right] C. \text { Gravitational potential } III. \left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right] D. \text { Acceleration due to gravity } IV. \left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]$ Choose the correct answer from the options given below:

Options:

A) A-IV, B-III, C-II, D-I

B) A-II, B-IV, C-III, D-I

C) A-I, B-III, C-IV, D-II

D) A-III, B-II, C-I, D-IV

EasyJEE Mains2025

Earth has mass 8 times and radius 2 times that of a planet. If the escape velocity from the earth is 11.2 km/s, the escape velocity in km/s from the planet will be:

Options:

A) 8.4

B) 11.2

C) 5.6

D) 2.8

MediumJEE Mains2025

A satellite is launched into a circular orbit of radius ' R ' around the earth. A second satellite is launched into an orbit of radius 1.03 R . The time period of revolution of the second satellite is larger than the first one approximately by

Options:

A) 3 \%

B) 2.5 \%

C) 4.5 \%

D) 9 \%

EasyJEE Mains2025

If a satellite orbiting the Earth is 9 times closer to the Earth than the Moon, what is the time period of rotation of the satellite? Given rotational time period of Moon =27 days and gravitational attraction between the satellite and the moon is neglected.

Options:

A) 3 days

B) 27 days

C) 81 days

D) 1 day

MediumJEE Mains2025

A small point of mass m is placed at a distance 2 R from the centre ' O ' of a big uniform solid sphere of mass M and radius R . The gravitational force on ' m ' due to M is \mathrm{F}_1. A spherical part of radius \mathrm{R} / 3 is removed from the big sphere as shown in the figure and the gravitational force on m due to remaining part of M is found to be F_2. The value of ratio F_1: F_2 is

Options:

A) 11 : 10

B) 12 : 11

C) 16 : 9

D) 12 : 9

MediumJEE Mains2024

A satellite of $10^3 \mathrm{~kg} mass is revolving in circular orbit of radius 2 R. If \frac{10^4 R}{6} \mathrm{~J} energy is supplied to the satellite, it would revolve in a new circular orbit of radius (use g=10 \mathrm{~m} / \mathrm{s}^2, R=$ radius of earth)

Options:

A) 4 R

B) 6 R

C) 2.5 R

D) 3 R

MediumJEE Mains2024

An astronaut takes a ball of mass $m from earth to space. He throws the ball into a circular orbit about earth at an altitude of 318.5 \mathrm{~km}. From earth's surface to the orbit, the change in total mechanical energy of the ball is x \frac{\mathrm{GM}_{\mathrm{e}} \mathrm{m}}{21 \mathrm{R}_{\mathrm{e}}}. The value of x is (take \mathrm{R}_{\mathrm{e}}=6370 \mathrm{~km})$ :

Options:

A) 12

B) 11

C) 9

D) 10

EasyJEE Mains2024

Two satellite A and B go round a planet in circular orbits having radii 4R and R respectively. If the speed of $\mathrm{A} is 3 v, the speed of \mathrm{B}$ will be :

Options:

A) 6 v

B) \frac{4}{3} v

C) 3 v

D) 12 v

MediumJEE Mains2024

Two planets $A and B having masses m_1 and m_2 move around the sun in circular orbits of r_1 and r_2 radii respectively. If angular momentum of A is L and that of B is 3 \mathrm{~L}, the ratio of time period \left(\frac{T_A}{T_B}\right)$ is:

Options:

A) \left(\frac{r_2}{r_1}\right)^{\frac{3}{2}}

B) 27\left(\frac{m_1}{m_2}\right)^3

C) \left(\frac{r_1}{r_2}\right)^3

D) \frac{1}{27}\left(\frac{m_2}{m_1}\right)^3

EasyJEE Mains2024

Assuming the earth to be a sphere of uniform mass density, a body weighed $300 \mathrm{~N}$ on the surface of earth. How much it would weigh at R/4 depth under surface of earth ?

Options:

A) 75 N

B) 375 N

C) 300 N

D) 225 N

EasyJEE Mains2024

To project a body of mass $m from earth's surface to infinity, the required kinetic energy is (assume, the radius of earth is R_E, g=$ acceleration due to gravity on the surface of earth):

Options:

A) 1 / 2 m g R_E

B) 4 m g R_E

C) m g R_E

D) 2 m g R_E

MediumJEE Mains2024

A satellite revolving around a planet in stationary orbit has time period 6 hours. The mass of planet is one-fourth the mass of earth. The radius orbit of planet is : (Given $= Radius of geo-stationary orbit for earth is 4.2 \times 10^4 \mathrm{~km}$)

Options:

A) 1.68 \times 10^5 \mathrm{~km}

B) 1.4 \times 10^4 \mathrm{~km}

C) 8.4 \times 10^4 \mathrm{~km}

D) 1.05 \times 10^4 \mathrm{~km}

MediumJEE Mains2024

If $\mathrm{G} be the gravitational constant and \mathrm{u} be the energy density then which of the following quantity have the dimensions as that of the \sqrt{\mathrm{uG}}$ :

Options:

A) Gravitational potential

B) pressure gradient per unit mass

C) Energy per unit mass

D) Force per unit mass

EasyJEE Mains2024

Match List I with List II : LIST I LIST II A. Kinetic energy of planet I. $-\mathrm{GMm} / \mathrm{a} B. Gravitation Potential energy of sun-planet system II. \mathrm{GMm} / 2 \mathrm{a} C. Total mechanical energy of planet III. \frac{\mathrm{Gm}}{\mathrm{r}} D. Escape energy at the surface of planet for unit mass object IV. -\mathrm{GMm} / 2 \mathrm{a} (Where \mathrm{a}= radius of planet orbit, \mathrm{r}= radius of planet, \mathrm{M}= mass of Sun, \mathrm{m}=$ mass of planet) Choose the correct answer from the options given below :

Options:

A) (A)-(II), (B)-(I), (C)-(IV), (D)-(III)

B) (A)-(I), (B)-(II), (C)-(III), (D)-(IV)

C) (A)-(III), (B)-(IV), (C)-(I), (D)-(II)

D) (A)-(I), (B)-(IV), (C)-(II), (D)-(III)

EasyJEE Mains2024

A $90 \mathrm{~kg} body placed at 2 \mathrm{R} distance from surface of earth experiences gravitational pull of : (\mathrm{R}= Radius of earth, \mathrm{g}=10 \mathrm{~m} \mathrm{~s}^{-2}$)

Options:

A) 300 N

B) 225 N

C) 100 N

D) 120 N

EasyJEE Mains2024

Correct formula for height of a satellite from earths surface is :

Options:

A) \left(\frac{T^2 R^2 g}{4 \pi^2}\right)^{1 / 3}-R

B) \left(\frac{T^2 R^2 g}{4 \pi}\right)^{1 / 2}-R

C) \left(\frac{T^2 R^2 g}{4 \pi^2}\right)^{-1 / 3}+R

D) \left(\frac{T^2 R^2}{4 \pi^2 g}\right)^{1 / 3}-R

MediumJEE Mains2024

A metal wire of uniform mass density having length $L and mass M is bent to form a semicircular arc and a particle of mass \mathrm{m}$ is placed at the centre of the arc. The gravitational force on the particle by the wire is :

Options:

A) \frac{\mathrm{GmM} \pi^2}{\mathrm{~L}^2}

B) \frac{\mathrm{GMm} \pi}{2 \mathrm{~L}^2}

C) 0

D) \frac{2 \mathrm{GmM} \pi}{\mathrm{L}^2}

MediumJEE Mains2024

A light planet is revolving around a massive star in a circular orbit of radius \mathrm{R} with a period of revolution T. If the force of attraction between planet and star is proportional to \mathrm{R}^{-3 / 2} then choose the correct option :

Options:

A) \mathrm{T}^2 \propto \mathrm{R}^{7 / 2}

B) \mathrm{T}^2 \propto \mathrm{R}^3

C) \mathrm{T}^2 \propto \mathrm{R}^{5 / 2}

D) \mathrm{T}^2 \propto \mathrm{R}^{3 / 2}

MediumJEE Mains2024

If \mathrm{R} is the radius of the earth and the acceleration due to gravity on the surface of earth is g=\pi^2 \mathrm{~m} / \mathrm{s}^2, then the length of the second's pendulum at a height \mathrm{h}=2 R from the surface of earth will be, :

Options:

A) \frac{1}{9} \mathrm{~m}

B) \frac{8}{9} \mathrm{~m}

C) \frac{2}{9} \mathrm{~m}

D) \frac{4}{9} \mathrm{~m}

EasyJEE Mains2024

The mass of the moon is $\frac{1}{144} times the mass of a planet and its diameter is \frac{1}{16} times the diameter of a planet. If the escape velocity on the planet is v$, the escape velocity on the moon will be :

Options:

A) \frac{\mathrm{v}}{4}

B) \frac{\mathrm{v}}{6}

C) \frac{\mathrm{V}}{12}

D) \frac{\mathrm{v}}{3}

MediumJEE Mains2024

Four identical particles of mass $m are kept at the four corners of a square. If the gravitational force exerted on one of the masses by the other masses is \left(\frac{2 \sqrt{2}+1}{32}\right) \frac{\mathrm{Gm}^2}{L^2}$, the length of the sides of the square is

Options:

A) 4L

B) 3L

C) 2L

D) \frac{L}{2}

EasyJEE Mains2024

Escape velocity of a body from earth is $11.2 \mathrm{~km} / \mathrm{s}$. If the radius of a planet be onethird the radius of earth and mass be one-sixth that of earth, the escape velocity from the planet is :

Options:

A) 7.9 km/s

B) 8.4 km/s

C) 4.2 km/s

D) 11.2 km/s

MediumJEE Mains2024

The gravitational potential at a point above the surface of earth is $-5.12 \times 10^7 \mathrm{~J} / \mathrm{kg} and the acceleration due to gravity at that point is 6.4 \mathrm{~m} / \mathrm{s}^2. Assume that the mean radius of earth to be 6400 \mathrm{~km}$. The height of this point above the earth's surface is :

Options:

A) 1600 km

B) 1200 km

C) 540 km

D) 1000 km

EasyJEE Mains2024

A planet takes 200 days to complete one revolution around the Sun. If the distance of the planet from Sun is reduced to one fourth of the original distance, how many days will it take to complete one revolution :

Options:

A) 20

B) 50

C) 100

D) 25

HardJEE Mains2024

At what distance above and below the surface of the earth a body will have same weight. (take radius of earth as $R$.)

Options:

A) \frac{\sqrt{3} R-R}{2}

B) \frac{R}{2}

C) \frac{\sqrt{5} R-R}{2}

D) \sqrt{5} R-R

EasyJEE Mains2024

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : The angular speed of the moon in its orbit about the earth is more than the angular speed of the earth in its orbit about the sun. Reason (R) : The moon takes less time to move around the earth than the time taken by the earth to move around the sun. In the light of the above statements, choose the most appropriate answer from the options given below :

Options:

A) Both (A) and (R) are correct but (R) is not the correct explanation of (A)

B) (A) is correct but (R) is not correct

C) Both (A) and (R) are correct and (R) is the correct explanation of (A)

D) (A) is not correct but (R) is correct

EasyJEE Mains2024

The acceleration due to gravity on the surface of earth is $\mathrm{g}$. If the diameter of earth reduces to half of its original value and mass remains constant, then acceleration due to gravity on the surface of earth would be :

Options:

A) g/4

B) 2g

C) g/2

D) 4g

EasyJEE Mains2023

Two identical particles each of mass ' m ' go round a circle of radius a under the action of their mutual gravitational attraction. The angular speed of each particle will be :

Options:

A) \sqrt{\frac{G m}{2 a^{3}}}

B) \sqrt{\frac{G m}{a^{3}}}

C) \sqrt{\frac{G m}{8 a^{3}}}

D) \sqrt{\frac{G m}{4 a^{3}}}

MediumJEE Mains2023

A body is released from a height equal to the radius (\mathrm{R}) of the earth. The velocity of the body when it strikes the surface of the earth will be (Given g= acceleration due to gravity on the earth.)

Options:

A) \sqrt{\frac{g R}{2}}

B) \sqrt{4 g R}

C) \sqrt{2 g R}

D) \sqrt{g R}

100

EasyJEE Mains2023

Given below are two statements: Statement I : For a planet, if the ratio of mass of the planet to its radius increases, the escape velocity from the planet also increases. Statement II : Escape velocity is independent of the radius of the planet. In the light of above statements, choose the most appropriate answer form the options given below

Options:

A) Both Statement I and Statement II are correct

B) Statement I is correct but statement II is incorrect

C) Both Statement I and Statement II are incorrect

D) Statement I is incorrect but statement II is correct

101

EasyJEE Mains2023

Two planets A and B of radii $\mathrm{R} and 1.5 R have densities \rho and \rho / 2 respectively. The ratio of acceleration due to gravity at the surface of \mathrm{B} to \mathrm{A}$ is:

Options:

A) 2 : 1

B) 2 : 3

C) 4 : 3

D) 3 : 4

102

EasyJEE Mains2023

A planet having mass $9 \mathrm{Me} and radius 4 \mathrm{R}_{\mathrm{e}}, where \mathrm{Me} and \mathrm{Re} are mass and radius of earth respectively, has escape velocity in \mathrm{km} / \mathrm{s} given by: (Given escape velocity on earth \mathrm{V}_{\mathrm{e}}=11.2 \times 10^{3} \mathrm{~m} / \mathrm{s}$ )

Options:

A) 33.6

B) 11.2

C) 16.8

D) 67.2

103

EasyJEE Mains2023

The ratio of escape velocity of a planet to the escape velocity of earth will be:- Given: Mass of the planet is 16 times mass of earth and radius of the planet is 4 times the radius of earth.

Options:

A) 1: 4

B) 1: \sqrt{2}

C) 4: 1

D) 2: 1

104

EasyJEE Mains2023

Two satellites $\mathrm{A} and \mathrm{B} move round the earth in the same orbit. The mass of \mathrm{A} is twice the mass of \mathrm{B}$. The quantity which is same for the two satellites will be

Options:

A) Potential energy

B) Kinetic energy

C) Total energy

D) Speed

105

MediumJEE Mains2023

A space ship of mass $2 \times 10^{4} \mathrm{~kg} is launched into a circular orbit close to the earth surface. The additional velocity to be imparted to the space ship in the orbit to overcome the gravitational pull will be (if g=10 \mathrm{~m} / \mathrm{s}^{2} and radius of earth =6400 \mathrm{~km}$ ):

Options:

A) 7.9(\sqrt{2}-1) \mathrm{km} / \mathrm{s}

B) 11.2(\sqrt{2}-1) \mathrm{km} / \mathrm{s}

C) 7.4(\sqrt{2}-1) \mathrm{km} / \mathrm{s}

D) 8(\sqrt{2}-1) \mathrm{km} / \mathrm{s}

106

EasyJEE Mains2023

If $\mathrm{V}$ is the gravitational potential due to sphere of uniform density on it's surface, then it's value at the center of sphere will be:-

Options:

A) \frac{3 \mathrm{~V}}{2}

B) \frac{\mathrm{V}}{2}

C) \frac{4}{3} \mathrm{~V}

D) \mathrm{V}

107

EasyJEE Mains2023

The radii of two planets 'A' and 'B' are 'R' and '4R' and their densities are $\rho and \rho / 3 respectively. The ratio of acceleration due to gravity at their surfaces \left(g_{A}: g_{B}\right)$ will be:

Options:

A) 3 : 16

B) 4 : 3

C) 1 : 16

D) 3 : 4

108

EasyJEE Mains2023

The time period of a satellite, revolving above earth's surface at a height equal to $\mathrm{R} will be (Given g=\pi^{2} \mathrm{~m} / \mathrm{s}^{2}, \mathrm{R}=$ radius of earth)

Options:

A) \sqrt{32 R}

B) \sqrt{4 \mathrm{R}}

C) \sqrt{8 R}

D) \sqrt{2 R}

109

EasyJEE Mains2023

Given below are two statements: Statement I : Rotation of the earth shows effect on the value of acceleration due to gravity (g) Statement II : The effect of rotation of the earth on the value of 'g' at the equator is minimum and that at the pole is maximum. In the light of the above statements, choose the correct answer from the options given below

Options:

A) Statement I is false but statement II is true

B) Statement I is true but statement II is false

C) Both Statement I and Statement II are true

D) Both Statement I and Statement II are false

110

EasyJEE Mains2023

Two satellites of masses m and 3m revolve around the earth in circular orbits of radii r & 3r respectively. The ratio of orbital speeds of the satellites respectively is

Options:

A) 3 : 1

B) \sqrt3$ : 1

C) 1 : 1

D) 9 : 1

111

MediumJEE Mains2023

Assuming the earth to be a sphere of uniform mass density, the weight of a body at a depth $d=\frac{R}{2}$ from the surface of earth, if its weight on the surface of earth is 200 N, will be: (Given R = radius of earth)

Options:

A) 100 N

B) 400 N

C) 300 N

D) 500 N

112

EasyJEE Mains2023

The acceleration due to gravity at height $h above the earth if h << \mathrm{R}$ (Radius of earth) is given by

Options:

A) g^{\prime}=g\left(1-\frac{2 h}{R}\right)

B) g^{\prime}=g\left(1-\frac{2 h^{2}}{R^{2}}\right)

C) g^{\prime}=g\left(1-\frac{h^{2}}{2 R^{2}}\right)

D) g^{\prime}=g\left(1-\frac{h}{2 R}\right)

113

EasyJEE Mains2023

The orbital angular momentum of a satellite is L, when it is revolving in a circular orbit at height h from earth surface. If the distance of satellite from the earth centre is increased by eight times to its initial value, then the new angular momentum will be -

Options:

A) 9L

B) 8L

C) 4L

D) 3L

114

MediumJEE Mains2023

Given below are two statements: Statement I: If $\mathrm{E} be the total energy of a satellite moving around the earth, then its potential energy will be \frac{E}{2}. Statement II: The kinetic energy of a satellite revolving in an orbit is equal to the half the magnitude of total energy \mathrm{E}$. In the light of the above statements, choose the most appropriate answer from the options given below

Options:

A) Both Statement I and Statement II are incorrect

B) Statement I is incorrect but Statement II is correct

C) Statement I is correct but Statement II is incorrect

D) Both Statement I and Statement II are correct

115

EasyJEE Mains2023

The weight of a body on the earth is $400 \mathrm{~N}$. Then weight of the body when taken to a depth half of the radius of the earth will be:

Options:

A) 300 N

B) 200 N

C) 100 N

D) Zero

116

EasyJEE Mains2023

The weight of a body on the surface of the earth is $100 \mathrm{~N}$. The gravitational force on it when taken at a height, from the surface of earth, equal to one-fourth the radius of the earth is:

Options:

A) 50 N

B) 64 N

C) 25 N

D) 100 N

117

EasyJEE Mains2023

Choose the incorrect statement from the following:

Options:

A) The linear speed of a planet revolving around the sun remains constant.

B) When a body falls towards earth, the displacement of earth towards the body is negligible.

C) The speed of satellite in a given circular orbit remains constant.

D) For a planet revolving around the sun in an elliptical orbit, the total energy of the planet remains constant.

118

EasyJEE Mains2023

Given below are two statements : one is labelled as Assertion $\mathbf{A} and the other is labelled as Reason \mathbf{R}$. Assertion A : Earth has atmosphere whereas moon doesn't have any atmosphere. Reason R : The escape velocity on moon is very small as compared to that on earth. In the light of the above statements, choose the correct answer from the options given below:

Options:

A) \mathbf{A} is false but \mathbf{R}$ is true

B) Both $\mathbf{A} and \mathbf{R} are correct but \mathbf{R} is NOT the correct explanation of \mathbf{A}

C) Both $\mathbf{A} and \mathbf{R} are correct and \mathbf{R} is the correct explanation of \mathbf{A}

D) \mathbf{A} is true but \mathbf{R}$ is false

119

MediumJEE Mains2023

A planet has double the mass of the earth. Its average density is equal to that of the earth. An object weighing $\mathrm{W}$ on earth will weigh on that planet:

Options:

A) 2^{2 / 3} \mathrm{~W}

B) W

C) 2 \mathrm{~W}

D) 2^{1 / 3} \mathrm{~W}

120

MediumJEE Mains2023

The escape velocities of two planets $\mathrm{A} and \mathrm{B} are in the ratio 1: 2. If the ratio of their radii respectively is 1: 3$, then the ratio of acceleration due to gravity of planet A to the acceleration of gravity of planet B will be :

Options:

A) \frac{4}{3}

B) \frac{2}{3}

C) \frac{3}{4}

D) \frac{3}{2}

121

EasyJEE Mains2023

For a body projected at an angle with the horizontal from the ground, choose the correct statement.

Options:

A) Gravitational potential energy is maximum at the highest point.

B) The vertical component of momentum is maximum at the highest point.

C) The horizontal component of velocity is zero at the highest point.

D) The Kinetic Energy (K.E.) is zero at the highest point of projectile motion.

122

EasyJEE Mains2023

If earth has a mass nine times and radius twice to that of a planet P. Then $\frac{v_{e}}{3} \sqrt{x} \mathrm{~ms}^{-1} will be the minimum velocity required by a rocket to pull out of gravitational force of \mathrm{P}, where v_{e} is escape velocity on earth. The value of x$ is

Options:

A) 1

B) 3

C) 2

D) 18

123

EasyJEE Mains2023

Given below are two statements: Statement I: Acceleration due to gravity is different at different places on the surface of earth. Statement II: Acceleration due to gravity increases as we go down below the earth's surface. In the light of the above statements, choose the correct answer from the options given below

Options:

A) Both Statement I and Statement II are true

B) Both Statement I and Statement II are false

C) Statement I is false but Statement II is true

D) Statement I is true but Statement II is false

124

EasyJEE Mains2023

A body weight \mathrm{W}, is projected vertically upwards from earth's surface to reach a height above the earth which is equal to nine times the radius of earth. The weight of the body at that height will be :

Options:

A) \frac{W}{91}

B) \frac{\mathrm{W}}{3}

C) \frac{\mathrm{W}}{100}

D) \frac{\mathrm{W}}{9}

125

MediumJEE Mains2023

At a certain depth "d " below surface of earth, value of acceleration due to gravity becomes four times that of its value at a height $\mathrm{3 R} above earth surface. Where \mathrm{R} is Radius of earth (Take \mathrm{R}=6400 \mathrm{~km} ). The depth \mathrm{d}$ is equal to

Options:

A) 5260 km

B) 2560 km

C) 640 km

D) 4800 km

126

MediumJEE Mains2023

An object is allowed to fall from a height R above the earth, where R is the radius of earth. Its velocity when it strikes the earth's surface, ignoring air resistance, will be

Options:

A) \sqrt{\frac{g R}{2}}

B) \sqrt{g R}

C) \sqrt{2 g R}

D) 2 \sqrt{g R}

127

MediumJEE Mains2023

If the gravitational field in the space is given as $\left(-\frac{K}{r^{2}}\right). Taking the reference point to be at \mathrm{r}=2 \mathrm{~cm} with gravitational potential \mathrm{V}=10 \mathrm{~J} / \mathrm{kg}. Find the gravitational potential at \mathrm{r}=3 \mathrm{~cm} in SI unit (Given, that \mathrm{K}=6 \mathrm{~Jcm} / \mathrm{kg}$)

Options:

A) 9

B) 11

C) 10

D) 12

128

EasyJEE Mains2023

The time period of a satellite of earth is 24 hours. If the separation between the earth and the satellite is decreased to one fourth of the previous value, then its new time period will become.

Options:

A) 12 hours

B) 3 hours

C) 6 hours

D) 4 hours

129

MediumJEE Mains2023

Two particles of equal mass '$m' move in a circle of radius 'r$' under the action of their mutual gravitational attraction. The speed of each particle will be :

Options:

A) \sqrt{\frac{G m}{4 r}}

B) \sqrt{\frac{G m}{2 r}}

C) \sqrt{\frac{G m}{r}}

D) \sqrt{\frac{4 G m}{r}}

130

EasyJEE Mains2023

Every planet revolves around the sun in an elliptical orbit :- A. The force acting on a planet is inversely proportional to square of distance from sun. B. Force acting on planet is inversely proportional to product of the masses of the planet and the sun. C. The Centripetal force acting on the planet is directed away from the sun. D. The square of time period of revolution of planet around sun is directly proportional to cube of semi-major axis of elliptical orbit. Choose the correct answer from the options given below :

Options:

A) C and D only

B) B and C only

C) A and D only

D) A and C only

131

EasyJEE Mains2023

A body of mass is taken from earth surface to the height h equal to twice the radius of earth (R$_e$), the increase in potential energy will be : (g = acceleration due to gravity on the surface of Earth)

Options:

A) \frac{1}{2}mgR_e

B) 3~mgR_e

C) \frac{1}{3}mgR_e

D) \frac{2}{3}mgR_e

132

MediumJEE Mains2023

Assume that the earth is a solid sphere of uniform density and a tunnel is dug along its diameter throughout the earth. It is found that when a particle is released in this tunnel, it executes a simple harmonic motion. The mass of the particle is 100 g. The time period of the motion of the particle will be (approximately) (Take g = 10 m s$^{-2}$ , radius of earth = 6400 km)

Options:

A) 12 hours

B) 1 hour 24 minutes

C) 24 hours

D) 1 hour 40 minutes

133

MediumJEE Mains2023

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R. Assertion A : A pendulum clock when taken to Mount Everest becomes fast. Reason R : The value of g (acceleration due to gravity) is less at Mount Everest than its value on the surface of earth. In the light of the above statements, choose the most appropriate answer from the options given below

Options:

A) Both A and R are correct but R is NOT the correct explanation of A

B) A is correct but R is not correct

C) Both A and R are correct and R is the correct explanation of A

D) A is not correct but R is correct

134

EasyJEE Mains2023

If the distance of the earth from Sun is 1.5 $\times 10^6$ km. Then the distance of an imaginary planet from Sun, if its period of revolution is 2.83 years is :

Options:

A) 6\times10^6$ km

B) 3\times10^7$ km

C) 6\times10^7$ km

D) 3\times10^6$ km

135

MediumJEE Mains2023

Given below are two statements: Statement I : Acceleration due to earth's gravity decreases as you go 'up' or 'down' from earth's surface. Statement II : Acceleration due to earth's gravity is same at a height 'h' and depth 'd' from earth's surface, if h = d. In the light of above statements, choose the most appropriate answer from the options given below

Options:

A) Statement I is correct but statement II is incorect

B) Both Statement I and II are correct

C) Statement I is incorrect but statement II is correct

D) Both Statement I and Statement II are incorrect

136

EasyJEE Mains2023

The weight of a body at the surface of earth is 18 N. The weight of the body at an altitude of 3200 km above the earth's surface is (given, radius of earth $\mathrm{R_e=6400~km}$) :

Options:

A) 9.8 N

B) 4.9 N

C) 19.6 N

D) 8 N

137

MediumJEE Mains2022

An object of mass $1 \mathrm{~kg} is taken to a height from the surface of earth which is equal to three times the radius of earth. The gain in potential energy of the object will be [If, \mathrm{g}=10 \mathrm{~ms}^{-2} and radius of earth =6400 \mathrm{~km}$ ]

Options:

A) 48 MJ

B) 24 MJ

C) 36 MJ

D) 12 MJ

138

EasyJEE Mains2022

If the radius of earth shrinks by $2 \%$ while its mass remains same. The acceleration due to gravity on the earth's surface will approximately :

Options:

A) decrease by $2 \%

B) decrease by $4 \%

C) increase by $2 \%

D) increase by $4 \%

139

MediumJEE Mains2022

A body of mass $\mathrm{m} is projected with velocity \lambda \,v_{\mathrm{e}} in vertically upward direction from the surface of the earth into space. It is given that v_{\mathrm{e}} is escape velocity and \lambda<1$. If air resistance is considered to be negligible, then the maximum height from the centre of earth, to which the body can go, will be : (R : radius of earth)

Options:

A) \frac{\mathrm{R}}{1+\lambda^{2}}

B) \frac{R}{1-\lambda^{2}}

C) \frac{R}{1-\lambda}

D) \frac{\lambda^{2} \mathrm{R}}{1-\lambda^{2}}

140

EasyJEE Mains2022

Two satellites $\mathrm{A} and \mathrm{B}, having masses in the ratio 4: 3, are revolving in circular orbits of radii 3 \mathrm{r} and 4 \mathrm{r} respectively around the earth. The ratio of total mechanical energy of \mathrm{A} to \mathrm{B}$ is :

Options:

A) 9 : 16

B) 16 : 9

C) 1 : 1

D) 4 : 3

141

MediumJEE Mains2022

A body is projected vertically upwards from the surface of earth with a velocity equal to one third of escape velocity. The maximum height attained by the body will be : (Take radius of earth $=6400 \mathrm{~km} and \mathrm{g}=10 \mathrm{~ms}^{-2}$ )

Options:

A) 800 km

B) 1600 km

C) 2133 km

D) 4800 km

142

EasyJEE Mains2022

The percentage decrease in the weight of a rocket, when taken to a height of $32 \mathrm{~km} above the surface of earth will, be : ( Radius of earth =6400 \mathrm{~km})

Options:

A) 1%

B) 3%

C) 4%

D) 0.5%

143

EasyJEE Mains2022

The length of a seconds pendulum at a height h = 2R from earth surface will be: (Given R = Radius of earth and acceleration due to gravity at the surface of earth, g = $\pi2 ms-$2)

Options:

A) {2 \over 9}$ m

B) {4 \over 9}$ m

C) {8 \over 9}$ m

D) {1 \over 9}$ m

144

EasyJEE Mains2022

An object is taken to a height above the surface of earth at a distance ${5 \over 4}$ R from the centre of the earth. Where radius of earth, R = 6400 km. The percentage decrease in the weight of the object will be :

Options:

A) 36%

B) 50%

C) 64%

D) 25%

145

MediumJEE Mains2022

Three identical particles $\mathrm{A}, \mathrm{B} and \mathrm{C} of mass 100 \mathrm{~kg} each are placed in a straight line with \mathrm{AB}=\mathrm{BC}=13 \mathrm{~m}. The gravitational force on a fourth particle \mathrm{P} of the same mass is \mathrm{F}, when placed at a distance 13 \mathrm{~m} from the particle \mathrm{B} on the perpendicular bisector of the line \mathrm{AC}. The value of \mathrm{F}$ will be approximately :

Options:

A) 21 G

B) 100 G

C) 59 G

D) 42 G

146

EasyJEE Mains2022

The radii of two planets A and B are in the ratio 2 : 3. Their densities are 3$\rho and 5\rho$ respectively. The ratio of their acceleration due to gravity is :

Options:

A) 9 : 4

B) 9 : 8

C) 9 : 10

D) 2 : 5

147

EasyJEE Mains2022

The time period of a satellite revolving around earth in a given orbit is 7 hours. If the radius of orbit is increased to three times its previous value, then approximate new time period of the satellite will be

Options:

A) 40 hours

B) 36 hours

C) 30 hours

D) 25 hours

148

MediumJEE Mains2022

The escape velocity of a body on a planet 'A' is 12 kms$-$1. The escape velocity of the body on another planet 'B', whose density is four times and radius is half of the planet 'A', is :

Options:

A) 12 kms$-$1

B) 24 kms$-$1

C) 36 kms$-$1

D) 6 kms$-$1

149

EasyJEE Mains2022

Water falls from a 40 m high dam at the rate of 9 $\times 104 kg per hour. Fifty percentage of gravitational potential energy can be converted into electrical energy. Using this hydroelectric energy number of 100 W lamps, that can be lit, is : (Take g = 10 ms-$2)

Options:

A) 25

B) 50

C) 100

D) 18

150

EasyJEE Mains2022

Two objects of equal masses placed at certain distance from each other attracts each other with a force of F. If one-third mass of one object is transferred to the other object, then the new force will be :

Options:

A) {2 \over 9}$ F

B) {16 \over 9}$ F

C) {8 \over 9}$ F

D) F

151

EasyJEE Mains2022

Two planets A and B of equal mass are having their period of revolutions TA and TB such that TA = 2TB. These planets are revolving in the circular orbits of radii rA and rB respectively. Which out of the following would be the correct relationship of their orbits?

Options:

A) 2r_A^2 = r_B^3

B) r_A^3 = 2r_B^3

C) r_A^3 = 4r_B^3

D) T_A^2 - T_B^2 = {{{\pi ^2}} \over {GM}}\left( {r_B^3 - 4r_A^3} \right)

152

EasyJEE Mains2022

The distance of the Sun from earth is 1.5 $\times$ 1011 m and its angular diameter is (2000) s when observed from the earth. The diameter of the Sun will be :

Options:

A) 2.45 $\times$ 1010 m

B) 1.45 $\times$ 1010 m

C) 1.45 $\times$ 109 m

D) 0.14 $\times$ 109 m

153

MediumJEE Mains2022

Four spheres each of mass m from a square of side d (as shown in figure). A fifth sphere of mass M is situated at the centre of square. The total gravitational potential energy of the system is :

Options:

A) - {{Gm} \over d}\left[ {(4 + \sqrt 2 )m + 4\sqrt 2 M} \right]

B) - {{Gm} \over d}\left[ {(4 + \sqrt 2 )M + 4\sqrt 2 m} \right]

C) - {{Gm} \over d}\left[ {3{m^2} + 4\sqrt 2 M} \right]

D) - {{Gm} \over d}\left[ {6{m^2} + 4\sqrt 2 M} \right]

154

EasyJEE Mains2022

Given below are two statements : Statement I : The law of gravitation holds good for any pair of bodies in the universe. Statement II : The weight of any person becomes zero when the person is at the centre of the earth. In the light of the above statements, choose the correct answer from the options given below.

Options:

A) Both Statement I and Statement II are true

B) Both Statement I and Statement II are false

C) Statement I is true but Statement II is false

D) Statement I is false but Statement II is true

155

EasyJEE Mains2022

Given below are two statements : One is labelled as Assertion A and the other is labelled as Reason R. Assertion A : If we move from poles to equator, the direction of acceleration due to gravity of earth always points towards the center of earth without any variation in its magnitude. Reason R : At equator, the direction of acceleration due to the gravity is towards the center of earth. In the light of above statements, choose the correct answer from the options given below:

Options:

A) Both A and R are true and R is the correct explanation of A.

B) Both A and R are true but R is NOT the correct explanation of A.

C) A is true but R is false.

D) A is false but R is true.

156

EasyJEE Mains2022

The variation of acceleration due to gravity (g) with distance (r) from the center of the earth is correctly represented by : (Given R = radius of earth)

Options:

157

EasyJEE Mains2022

The height of any point P above the surface of earth is equal to diameter of earth. The value of acceleration due to gravity at point P will be : (Given g = acceleration due to gravity at the surface of earth).

Options:

A) g/2

B) g/4

C) g/3

D) g/9

158

EasyJEE Mains2022

The distance between Sun and Earth is R. The duration of year if the distance between Sun and Earth becomes 3R will be :

Options:

A) \sqrt 3 $ years

B) 3 years

C) 9 years

D) 3$\sqrt 3 $ years

159

EasyJEE Mains2022

The approximate height from the surface of earth at which the weight of the body becomes ${1 \over 3} of its weight on the surface of earth is : [Radius of earth R = 6400 km and \sqrt 3 $ = 1.732]

Options:

A) 3840 km

B) 4685 km

C) 2133 km

D) 4267 km

160

MediumJEE Mains2021

Four particles each of mass M, move along a circle of radius R under the action of their mutual gravitational attraction as shown in figure. The speed of each particle is :

Options:

A) {1 \over 2}\sqrt {{{GM} \over {R(2\sqrt 2 + 1)}}}

B) {1 \over 2}\sqrt {{{GM} \over R}(2\sqrt 2 + 1)}

C) {1 \over 2}\sqrt {{{GM} \over R}(2\sqrt 2 - 1)}

D) \sqrt {{{GM} \over R}}

161

MediumJEE Mains2021

If RE be the radius of Earth, then the ratio between the acceleration due to gravity at a depth 'r' below and a height 'r' above the earth surface is : (Given : r < RE)

Options:

A) 1 - {r \over {{R_E}}} - {{{r^2}} \over {R_E^2}} - {{{r^3}} \over {R_E^3}}

B) 1 + {r \over {{R_E}}} + {{{r^2}} \over {R_E^2}} + {{{r^3}} \over {R_E^3}}

C) 1 + {r \over {{R_E}}} - {{{r^2}} \over {R_E^2}} + {{{r^3}} \over {R_E^3}}

D) 1 + {r \over {{R_E}}} - {{{r^2}} \over {R_E^2}} - {{{r^3}} \over {R_E^3}}

162

MediumJEE Mains2021

The masses and radii of the earth and moon are (M1, R1) and (M2, R2) respectively. Their centres are at a distance 'r' apart. Find the minimum escape velocity for a particle of mass 'm' to be projected from the middle of these two masses :

Options:

A) V = {1 \over 2}\sqrt {{{4G({M_1} + {M_2})} \over r}}

B) V = \sqrt {{{4G({M_1} + {M_2})} \over r}}

C) V = {1 \over 2}\sqrt {{{2G({M_1} + {M_2})} \over r}}

D) V = {{\sqrt {2G} ({M_1} + {M_2})} \over r}

163

MediumJEE Mains2021

A mass of 50 kg is placed at the centre of a uniform spherical shell of mass 100 kg and radius 50 m. If the gravitational potential at a point, 25 m from the centre is V kg/m. The value of V is :

Options:

A) -$60 G

B) +2 G

C) -$20 G

D) -$4 G

164

EasyJEE Mains2021

Inside a uniform spherical shell :(1) the gravitational field is zero(2) the gravitational potential is zero(3) the gravitational field is same everywhere(4) the gravitational potential is same everywhere(5) all of the aboveChoose the most appropriate answer from the options given below :

Options:

A) (1), (3) and (4) only

B) (5) only

C) (1), (2) and (3) only

D) (2), (3) and (4) only

165

MediumJEE Mains2021

Two identical particles of mass 1 kg each go round a circle of radius R, under the action of their mutual gravitational attraction. The angular speed of each particle is :

Options:

A) \sqrt {{G \over {2{R^3}}}}

B) {1 \over 2}\sqrt {{G \over {{R^3}}}}

C) {1 \over {2R}}\sqrt {{1 \over G}}

D) {{2G} \over {{R^3}}}

166

EasyJEE Mains2021

The planet Mars has two moons, if one of them has a period 7 hours, 30 minutes and an orbital radius of 9.0 $\times 103 km. Find the mass of Mars.\left\{ {Given\,{{4{\pi ^2}} \over G} = 6 \times {{10}^{11}}{N^{ - 1}}{m^{ - 2}}k{g^2}} \right\}

Options:

A) 5.96 $\times$ 1019 kg

B) 3.25 $\times$ 1021 kg

C) 7.02 $\times$ 1025 kg

D) 6.00 $\times$ 1023 kg

167

MediumJEE Mains2021

Consider a planet in some solar system which has a mass double the mass of earth and density equal to the average density of earth. If the weight of an object on earth is W, the weight of the same object on that planet will be :

Options:

A) 2W

B) W

C) {2^{{1 \over 3}}}$W

D) \sqrt 2 $W

168

EasyJEE Mains2021

The minimum and maximum distances of a planet revolving around the sun are x1 and x2. If the minimum speed of the planet on its trajectory is v0 then its maximum speed will be :

Options:

A) {{{v_0}x_1^2} \over {x_2^2}}

B) {{{v_0}x_2^2} \over {x_1^2}}

C) {{{v_0}x_1^{}} \over {x_2^{}}}

D) {{{v_0}x_2^{}} \over {x_1^{}}}

169

HardJEE Mains2021

A body is projected vertically upwards from the surface of earth with a velocity sufficient enough to carry it to infinity. The time taken by it to reach height h is ___________ s.

Options:

A) \sqrt {{{2{R_e}} \over g}} \left[ {{{\left( {1 + {h \over {{R_e}}}} \right)}^{{3 \over 2}}} - 1} \right]

B) {1 \over 3}\sqrt {{{{R_e}} \over {2g}}} \left[ {{{\left( {1 + {h \over {{R_e}}}} \right)}^{{3 \over 2}}} - 1} \right]

C) \sqrt {{{{R_e}} \over {2g}}} \left[ {{{\left( {1 + {h \over {{R_e}}}} \right)}^{{3 \over 2}}} - 1} \right]

D) {1 \over 3}\sqrt {{{2{R_e}} \over g}} \left[ {{{\left( {1 + {h \over {{R_e}}}} \right)}^{{3 \over 2}}} - 1} \right]

170

EasyJEE Mains2021

A satellite is launched into a circular orbit of radius R around earth, while a second satellite is launched into a circular orbit of radius 1.02 R. The percentage difference in the time periods of the two satellites is :

Options:

A) 1.5

B) 2.0

C) 0.7

D) 3.0

171

EasyJEE Mains2021

Consider a binary star system of star A and star B with masses mA and mB revolving in a circular orbit of radii rA an rB, respectively. If TA and TB are the time period of star A and star B, respectively,Then :

Options:

A) {{{T_A}} \over {{T_B}}} = {\left( {{{{r_A}} \over {{r_B}}}} \right)^{{3 \over 2}}}

B) {T_A} = {T_B}

C) {T_A} > {T_B} (if {m_A} > {m_B}$)

D) {T_A} > {T_B} (if {r_A} > {r_B}$)

172

MediumJEE Mains2021

A person whose mass is 100 kg travels from Earth to Mars in a spaceship. Neglect all other objects in sky and take acceleration due to gravity on the surface of the Earth and Mars as 10 m/s2 and 4 m/s2 respectively. Identify from the below figures, the curve that fits best for the weight of the passenger as a function of time.

Options:

A) (b)

B) (c)

C) (d)

D) (a)

173

MediumJEE Mains2021

If the angular velocity of earth's spin is increased such that the bodies at the equator start floating, the duration of the day would be approximately : [Take g = 10 ms$-2, the radius of earth, R = 6400 \times 103 m, Take \pi$ = 3.14]

Options:

A) 84 minutes

B) 1200 minutes

C) 60 minutes

D) does not change

174

EasyJEE Mains2021

The angular momentum of a planet of mass M moving around the sun in an elliptical orbit is ${\overrightarrow L }$. The magnitude of the areal velocity of the planet is :

Options:

A) {{2L} \over M}

B) {{L} \over 2M}

C) {{L} \over M}

D) {{4L} \over M}

175

EasyJEE Mains2021

The time period of a satellite in a circular orbit of radius R is T. The period of another satellite in a circular orbit of radius 9R is :

Options:

A) 9 T

B) 27 T

C) 12 T

D) 3 T

176

MediumJEE Mains2021

A geostationary satellite is orbiting around an arbitrary planet 'P' at a height of 11R above the surface of 'P', R being the radius of 'P'. The time period of another satellite in hours at a height of 2R from the surface of 'P' is _________. 'P' has the time period of 24 hours.

Options:

A) 3

B) 5

C) 6\sqrt 2

D) {6 \over {\sqrt 2 }}

177

EasyJEE Mains2021

The maximum and minimum distances of a comet from the Sun are 1.6 $\times 1012 m and 8.0 \times 1010 m respectively. If the speed of the comet at the nearest point is 6 \times 104 ms-$1, the speed at the farthest point is :

Options:

A) 3.0 $\times$ 103 m/s

B) 6.0 $\times$ 103 m/s

C) 1.5 $\times$ 103 m/s

D) 4.5 $\times$ 103 m/s

178

EasyJEE Mains2021

A planet revolving in elliptical orbit has :A. a constant velocity of revolution.B. has the least velocity when it is nearest to the sun.C. its areal velocity is directly proportional to its velocity.D. areal velocity is inversely proportional to its velocity.E. to follow a trajectory such that the areal velocity is constant.Choose the correct answer from the options given below :

Options:

A) D only

B) E only

C) C only

D) A only

179

MediumJEE Mains2021

Find the gravitational force of attraction between the ring and sphere as shown in the diagram, where the plane of the ring is perpendicular to the line joining the centres. If $\sqrt 8 $R is the distance between the centres of a ring (of mass 'm') and a sphere (mass 'M') where both have equal radius 'R'.

Options:

A) {{2\sqrt 2 } \over 3}.{{GMm} \over {{R^2}}}

B) {{\sqrt 8 } \over 9}.{{GmM} \over R}

C) {{\sqrt 8 } \over {27}}.{{GmM} \over {{R^2}}}

D) {1 \over {3\sqrt 8 }}.{{GMm} \over {{R^2}}}

180

MediumJEE Mains2021

A solid sphere of radius R gravitationally attracts a particle placed at 3R from its centre with a force F1. Now a spherical cavity of radius $\left( {{R \over 2}} \right)$ is made in the sphere (as shown in figure) and the force becomes F2. The value of F1 : F2 is

Options:

A) 36 : 25

B) 41 : 50

C) 50 : 41

D) 25 : 36

181

MediumJEE Mains2021

Two satellites A and B of masses 200 kg and 400 kg are revolving round the earth at height of 600 km and 1600 km respectively.If TA and TB are the time periods of A and B respectively then the value of TB $- TA :[Given : radius of earth = 6400 km, mass of earth = 6 \times$ 1024 kg]

Options:

A) 1.33 $\times$ 103 s

B) 4.24 $\times$ 102 s

C) 3.33 $\times$ 102 s

D) 4.24 $\times$ 103 s

182

MediumJEE Mains2021

Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R.Assertion A : The escape velocities of planet A and B are same. But A and B are of unequal mass.Reason R : The product of their mass and radius must be same. M1R1 = M2R2In the light of the above statements, choose the most appropriate answer from the options given below :

Options:

A) Both A and R are correct and R is the correct explanation of A

B) Both A and R are correct but R is NOT the correct explanation of A

C) A is correct but R is not correct

D) A is not correct but R is correct

183

EasyJEE Mains2021

A body weights 49N on a spring balance at the north pole. What will be its weight recorded on the same weighing machine, if it is shifted to the equator?[Use $g = {{GM} \over {{R^2}}} = 9.8 ms-$2 and radius of earth, R = 6400 km.]

Options:

A) 49 N

B) 49.83 N

C) 48.83 N

D) 49.17 N

184

MediumJEE Mains2021

Four identical particles of equal masses 1 kg made to move along the circumference of a circle of radius 1 m under the action of their own mutual gravitational attraction. The speed of each particle will be :

Options:

A) \sqrt {{G \over 2}(1 + 2\sqrt 2 )}

B) \sqrt {{G \over 2}(2\sqrt 2 - 1)}

C) \sqrt {G(1 + 2\sqrt 2 )}

D) {1\over2}\sqrt {G(1 + 2\sqrt 2 )}

185

EasyJEE Mains2021

Consider two satellites S1 and S2 with periods of revolution 1 hr. and 8 hr. respectively revolving around a planet in circular orbits. The ratio of angular velocity of satellite S1 to the angular velocity of satellite S2 is :

Options:

A) 1 : 4

B) 8 : 1

C) 2 : 1

D) 1 : 8

186

MediumJEE Mains2021

Two stars of masses m and 2m at a distance d rotate about their common centre of mass in free space. The period of revolution is :

Options:

A) {1 \over {2\pi }}\sqrt {{{{d^3}} \over {3Gm}}}

B) 2\pi \sqrt {{{3Gm} \over {{d^3}}}}

C) {1 \over {2\pi }}\sqrt {{{3Gm} \over {{d^3}}}}

D) 2\pi \sqrt {{{{d^3}} \over {3Gm}}}

187

MediumJEE Mains2020

Two planets have masses M and 16 M and their radii are $a and 2a, respectively. The separation between the centres of the planets is 10a$. A body of mass m is fired from the surface of the larger planet towards the smaller planet along the line joining their centres. For the body to be able to reach at the surface of smaller planet, the minimum firing speed needed is :

Options:

A) 2\sqrt {{{GM} \over a}}

B) \sqrt {{{G{M^2}} \over {ma}}}

C) {3 \over 2}\sqrt {{{5GM} \over a}}

D) 4\sqrt {{{GM} \over a}}

188

MediumJEE Mains2020

A satellite is in an elliptical orbit around a planet P. It is observed that the velocity of the satellite when it is farthest from the planet is 6 times less than that when it is closest to the planet. The ratio of distances between the satellite and the planet at closest and farthest points is:

Options:

A) 1 : 2

B) 1 : 3

C) 1 : 6

D) 3 : 4

189

MediumJEE Mains2020

The acceleration due to gravity on the earth’s surface at the poles is g and angular velocity of the earth about the axis passing through the pole is $\omega $. An object is weighed at the equator and at a height h above the poles by using a spring balance. If the weights are found to be same, then h is (h << R, where R is the radius of the earth)

Options:

A) {{{R^2}{\omega ^2}} \over {2g}}

B) {{{R^2}{\omega ^2}} \over g}

C) {{{R^2}{\omega ^2}} \over {8g}}

D) {{{R^2}{\omega ^2}} \over {4g}}

190

MediumJEE Mains2020

The value of the acceleration due to gravity is g1 at a height h = ${R \over 2} (R = radius of the earth) from the surface of the earth. It is again equal to g1 at a depth d below the surface of the earth. The ratio \left( {{d \over R}} \right)$ equals :

Options:

A) {5 \over 9}

B) {1 \over 9}

C) {7 \over 9}

D) {4 \over 9}

191

MediumJEE Mains2020

A body is moving in a low circular orbit about a planet of mass M and radius R. The radius of the orbit can be taken to be R itself. Then the ratio of the speed of this body in the orbit to the escape velocity from the planet is:

Options:

A) 2

B) 1

C) \sqrt 2

D) {1 \over {\sqrt 2 }}

192

MediumJEE Mains2020

On the x-axis and at a distance x from the origin, the gravitational field due a mass distribution is given by ${{Ax} \over {{{\left( {{x^2} + {a^2}} \right)}^{3/2}}}}$ in the x-direction. The magnitude of gravitational potential on the x-axis at a distance x, taking its value to be zero at infinity, is:

Options:

A) {A{{\left( {{x^2} + {a^2}} \right)}^{3/2}}}

B) {A{{\left( {{x^2} + {a^2}} \right)}^{1/2}}}

C) {A \over {{{\left( {{x^2} + {a^2}} \right)}^{1/2}}}}

D) {A \over {{{\left( {{x^2} + {a^2}} \right)}^{3/2}}}}

193

MediumJEE Mains2020

The mass density of a planet of radius R varies with the distance r from its centre as $\rho (r) = {\rho _0}\left( {1 - {{{r^2}} \over {{R^2}}}} \right)$. Then the gravitational field is maximum at :

Options:

A) r = {1 \over {\sqrt 3 }}R

B) r = R

C) r = \sqrt {{3 \over 4}} R

D) r = \sqrt {{5 \over 9}} R

194

MediumJEE Mains2020

A satellite is moving in a low nearly circular orbit around the earth. Its radius is roughly equal to that of the earth’s radius Re . By firing rockets attached to it, its speed is instantaneously increased in the direction of its motion so that it become $\sqrt {{3 \over 2}} $ times larger. Due to this the farthest distance from the centre of the earth that the satellite reaches is R. Value of R is :

Options:

A) 2Re

B) 3Re

C) 4Re

D) 2.5Re

195

MediumJEE Mains2020

The height ‘h’ at which the weight of a body will be the same as that at the same depth ‘h’ from the surface of the earth is (Radius of the earth is R and effect of the rotation of the earth is neglected)

Options:

A) {R \over 2}

B) {{\sqrt 5 R - R} \over 2}

C) {{\sqrt 3 R - R} \over 2}

D) {{\sqrt 5 } \over 2}R - R

196

MediumJEE Mains2020

The mass density of a spherical galaxy varies as ${K \over r}$ over a large distance ‘r’ from its centre. In that region, a small star is in a circular orbit of radius R. Then the period of revolution, T depends on R as :

Options:

A) T2 $ \propto $ R

B) T2 $ \propto $ R3

C) T $ \propto $ R

D) T2 $ \propto {1 \over {{R^3}}}

197

MediumJEE Mains2020

Planet A has mass M and radius R. Planet B has half the mass and half the radius of Planet A. If the escape velocities from the Planets A and B are vA and vB, respectively, then ${{{v_A}} \over {{v_B}}} = {n \over 4}$. The value of n is :

Options:

A) 1

B) 2

C) 4

D) 3

198

MediumJEE Mains2020

A body A of mass m is moving in a circular orbit of radius R about a planet. Another body B of mass ${m \over 2} collides with A with a velocity which is half \left( {{{\overrightarrow v } \over 2}} \right) the instantaneous velocity{\overrightarrow v }$ of A. The collision is completely inelastic. Then, the combined body :

Options:

A) starts moving in an elliptical orbit around the planet.

B) Falls vertically downwards towards the planet

C) Escapes from the Planet's Gravitational field.

D) continues to move in a circular orbit

199

MediumJEE Mains2020

Consider two solid spheres of radii R1 = 1m, R2 = 2m and masses M1 and M2, respectively. The gravitational field due to sphere (1) and (2) are shown. The value of ${{{M_1}} \over {{M_2}}}$ is :

Options:

A) {2 \over 3}

B) {1 \over 6}

C) {1 \over 2}

D) {1 \over 3}

200

MediumJEE Mains2020

A box weight 196 N on a spring balance at the north pole. Its weight recorded on the same balance if it is shifted to the equator is close to (Take g = 10 ms–2 at the north pole and the radius of the earth = 6400 km) :

Options:

A) 194.32 N

B) 195.66 N

C) 195.32 N

D) 194.66 N

201

MediumJEE Mains2020

A satellite of mass m is launched vertically upwards with an initial speed u from the surface of the earth. After it reaches height R (R = radius of the earth), it ejects a rocket of mass ${m \over {10}}$ so that subsequently the satellite moves in a circular orbit. The kinetic energy of the rocket is (G is the gravitational constant; M is the mass of the earth) :

Options:

A) {{3m} \over 8}{\left( {u + \sqrt {{{5GM} \over {6R}}} } \right)^2}

B) {m \over {20}}\left( {{u^2} + {{113} \over {100}}{{GM} \over R}} \right)

C) 5m\left( {{u^2} - {{119} \over {100}}{{GM} \over R}} \right)

D) {m \over {20}}{\left( {u - \sqrt {{{2GM} \over {3R}}} } \right)^2}

202

MediumJEE Mains2019

The ratio of the weights of a body on the Earth’s surface to that on the surface of a planets is 9 : 4. The mass of the planet is ${1 \over 9}$ th of that of the Earth. If 'R' is the radius of the Earth, what is the radius of the planet ? (Take the planets to have the same mass density)

Options:

A) {R \over 9}

B) {R \over 2}

C) {R \over 3}

D) {R \over 4}

203

MediumJEE Mains2019

A spaceship orbits around a planet at a height of 20 km from its surface. Assuming that only gravitational field of the planet acts on the spaceship, what will be the number of complete revolutions made by the spaceship in 24 hours around the planet? [Given ; Mass of planet = 8 × 1022 kg, Radius of planet = 2 × 106 m, Gravitational constant G = 6.67 × 10–11 Nm2 /kg2]

Options:

A) 13

B) 9

C) 17

D) 11

204

MediumJEE Mains2019

The value of acceleration due to gravity at Earth's surface is 9.8 ms–2. The altitude above its surface at which the acceleration due to gravity decreases to 4.9 ms–2, is close to : (Radius of earth = 6.4 × 106 m)

Options:

A) 1.6 × 106 m

B) 9.0 × 106 m

C) 6.4 × 106 m

D) 2.6 × 106 m

205

MediumJEE Mains2019

A test particle is moving in a circular orbit in the gravitational field produced by a mass density $\rho (r) = {K \over {{r^2}}}$ . Identify the correct relation between the radius R of the particle's orbit and its period T

Options:

A) T2/R3 is a constant

B) TR is a constant

C) T/R2 is a constant

D) T/R is a constant

206

MediumJEE Mains2019

A solid sphere of mass 'M' and radius 'a' is surrounded by a uniform concentric spherical shell of thickness 2a and mass 2M. The gravitational field at distance '3a' from the centre will be :

Options:

A) {{GM} \over {3{a^2}}}

B) {{2GM} \over {9{a^2}}}

C) {{GM} \over {9{a^2}}}

D) {{2GM} \over {3{a^2}}}

207

MediumJEE Mains2019

A rocket has to be launched from earth in such a way that it never returns. If E is the minimum energy delivered by the rocket launcher, what should be the minimum energy that the launcher should have if the same rocket is to be launched from the surface of the moon ? Assume that the density of the earth and the moon are equal and that the earth's volume is 64 times the volume of the moon :-

Options:

A) E/32

B) E/16

C) E/4

D) E/64

208

MediumJEE Mains2019

Four identical particles of mass M are located at the corners of a square of side 'a'. What should be their speed if each of them revolves under the influence of other's gravitational field in a circular orbit circumscribing the square?

Options:

A) 1.21\sqrt {{{GM} \over a}}

B) 1.16\sqrt {{{GM} \over a}}

C) 1.41\sqrt {{{GM} \over a}}

D) 1.35\sqrt {{{GM} \over a}}

209

MediumJEE Mains2019

Two satellites, A and B, have masses m and 2m respectively. A is in a circular orbit of radius R, and B is in a circular orbit of radius 2R around the earth. The ratio of their kinetic energies, TA/TB, is ;

Options:

A) 2

B) {{1 \over 2}}

C) \sqrt {{1 \over 2}}

D) 1

210

MediumJEE Mains2019

A satellite of mass M is in a circular orbit of radius R about the centre of the earth. A meteorite of the same mass, falling towards the earth, collides with the satellite completely inelastically. The speeds of the satellite and the meteorite are the same, just before the collision. The subsequent motion of the combined body will be :

Options:

A) in the same circular orbit of radius R

B) such that it escapes to infinity

C) in a circular orbit of a different radius

D) in an elliptical orbit

211

MediumJEE Mains2019

A straight rod of length L extends from x = a to x = L + a. The gravitational force it exerts on a point mass 'm' at x = 0, if the mass per unit length of the rod is A + Bx2 , is given by :

Options:

A) Gm\left[ {A\left( {{1 \over a} - {1 \over {a + L}}} \right) - BL} \right]

B) Gm\left[ {A\left( {{1 \over a} - {1 \over {a + L}}} \right) + BL} \right]

C) Gm\left[ {A\left( {{1 \over {a + L}} - {1 \over a}} \right) + BL} \right]

D) Gm\left[ {A\left( {{1 \over {a + L}} - {1 \over a}} \right) - BL} \right]

212

MediumJEE Mains2019

A satellite is revolving in a circular orbit at a height h form the earth surface, such that h < < R where R is the earth. Assuming that the effect of earth's atmosphere can be neglected the minimum increase in the speed required so that the satellite could escape from the gravitational field of earth is :

Options:

A) \sqrt {gR} \left( {\sqrt 2 - 1} \right)

B) \sqrt {2gR}

C) \sqrt {gR}

D) {{\sqrt {gR} } \over 2}

213

MediumJEE Mains2019

Two stars of masses 3 $ \times 1031 kg each, and at distance 2 \times 1011 m rotate in a plane about their common centre of mass O. A meteorite passes through O moving perpendicular to the star’s rotation plane. In order to escape from the gravitational field of this double star, the minimum speed that meteorite should have at O is - (Take Gravitational constant; G = 6.67 \times $ 10–11 Nm2 kg–2)

Options:

A) 2.4 $ \times $ 104 m/s

B) 1.4 $ \times $ 105 m/s

C) 3.8 $ \times $ 104 m/s

D) 2.8 $ \times $ 105 m/s

214

MediumJEE Mains2019

A satellite is moving with a constant speed v in circular orbit around the earth. An object of mass ‘m’ is ejected from the satellite such that it just escapes from the gravitational pull of the earth. At the time of ejection, the kinetic energy of the object is -

Options:

A) mv2

B) {1 \over 2}$ mv2

C) {3 \over 2}$ mv2

D) 2 mv2

215

MediumJEE Mains2019

The energy required to take a satellite to a height 'h' above Earth surface (radius of Earth = 6.4 $ \times $ 103 km) is E1 and kinetic energy required for the satellite to be in a circular orbit at this height is E2. The value of h for which E1 and E2 are equal, is

Options:

A) 1.6 $ \times $ 103 km

B) 3.2 $ \times $ 103 km

C) 6.4 $ \times $ 103 km

D) 1.28 $ \times $ 104 km

216

MediumJEE Mains2018

Suppose that the angular velocity of rotation of earth is increased. Then, as a consequence :

Options:

A) Weight of the object, everywhere on the earth, will increase.

B) Weight of the object, everywhere on the earth, will decrease.

C) There will be no change in weight anywhere on the earth.

D) Except at poles, weight of the object on the earth will decrease.

217

MediumJEE Mains2018

A particle is moving with a uniform speed in a circular orbit of radius R in a central force inversely proportional to the nth power of R. If the period of rotation of the particle is T, then :

Options:

A) T $ \propto $ Rn/2

B) T $ \propto $ R3/2 for any n

C) T $ \propto $ Rn/2 +1

D) T $ \propto $ R(n+1)/2

218

MediumJEE Mains2018

Take the mean distance of the moon and the sun from the earth to be $0.4 \times {10^6} km and 150 \times {10^6} km respectively. Their masses are 8 \times {10^{22}} kg and 2 \times {10^{30}} kg respectively. The radius of the earth is 6400 km. Let \Delta {F_1} be the difference in the forces exerted by the moon at the nearest and farthest points on the earth and \Delta {F_2} be the difference in the force exerted by the sun at the nearest and farthest points on the earth. Then, the number closest to {{\Delta {F_1}} \over {\Delta {F_2}}}$ is :

Options:

A) 2

B) {10^{ - 2}}

C) 0.6

D) 6

219

MediumJEE Mains2018

A body of mass m is moving in a circular orbit of radius R about a planet of mass M. At some instant, it splits into two equal masses. The first mass moves in a circular orbit of radius ${R \over 2}, and the other mass, in a circular orbit of radius {3R \over 2}$. The difference between the final and initial total energies is :

Options:

A) - {{GMm} \over {2R}}

B) + {{GMm} \over {6R}}

C) {{GMm} \over {2R}}

D) - {{GMm} \over {6R}}

220

MediumJEE Mains2017

The mass density of a spherical body is given by $\rho (r) = {k \over r} for r \le R and \rho $ (r) = 0 for r > R, where r is the distance from the centre. The correct graph that describes qualitatively the acceleration, a, of a test particle as a function of r is :

Options:

221

MediumJEE Mains2017

If the Earth has no rotational motion, the weight of a person on the equator is W. Determine the speed with which the earth would have to rotate about its axis so that the person at the equator will weigh ${3 \over 4}$ W. Radius of the Earth is 6400 km and g=10 m/s2.

Options:

A) 1.1 $ \times $ 10−3 rad/s

B) 0.83 $ \times $ 10−3 rad/s

C) 0.63 $ \times $ 10−3 rad/s

D) 0.28 $ \times $ 10−3 rad/s

222

MediumJEE Mains2017

The variation of acceleration due to gravity $g$ with distance d from centre of the earth is best represented by (R = Earth’s radius):

Options:

223

MediumJEE Mains2016

Figure shows elliptical path abcd of a planet around the sun S such that the area of triangle csa is ${1 \over 4}$ the area of the ellipse. (See figure) With db as the semimajor axis, and ca as the semiminor axis. If t1 is the time taken for planet to go over path abc and t2 for path taken over cda then :

Options:

A) t1 = t2

B) t1 = 2t2

C) t1 = 3t2

D) t1 = 4t2

224

MediumJEE Mains2016

A satellite is revolving in a circular orbit at a height $'h' from the earth's surface (radius of earth R;h < < R$). The minimum increase in its orbital velocity required, so that the satellite could escape from the earth's gravitational field, is close to : (Neglect the effect of atmosphere.)

Options:

A) \sqrt{2 g R}

B) \sqrt{g R}

C) \sqrt{g R / 2}

D) \sqrt{g R}(\sqrt{2}-1)

225

MediumJEE Mains2015

From a solid sphere of mass $M and radius R, a spherical portion of radius R/2 is removed, as shown in the figure. Taking gravitational potential V=0 at r = \infty , the potential at the center of the cavity thus formed is: (G=gravitational constant$)

Options:

A) {{ - 2GM} \over {3R}}

B) {{ - 2GM} \over R}

C) {{ - GM} \over {2R}}

D) {{ - GM} \over R}

226

MediumJEE Mains2014

Four particles, each of mass $M and equidistant from each other, move along a circle of radius R$ under the action of their mutual gravitational attraction. The speed of each particle is :

Options:

A) \sqrt {{{GM} \over R}}

B) \sqrt {2\sqrt 2 {{GM} \over R}}

C) \sqrt {{{GM} \over R}\left( {1 + 2\sqrt 2 } \right)}

D) {1 \over 2}\sqrt {{{GM} \over R}\left( {1 + 2\sqrt 2 } \right)}

227

MediumJEE Mains2013

What is the minimum energy required to launch a satellite of mass $m from the surface of a planet of mass M and radius R in a circular orbit at an altitude of 2R$?

Options:

A) {{5GmM} \over {6R}}

B) {{2GmM} \over {3R}}

C) {{GmM} \over {2R}}

D) {{GmM} \over {3R}}

228

MediumJEE Mains2012

The mass of a spaceship is $1000 kg. It is to be launched from the earth's surface out into free space. The value of g and R (radius of earth ) are 10\,m/{s^2} and 6400 km$ respectively. The required energy for this work will be:

Options:

A) 6.4 \times {10^{11}}\,$ Joules

B) 6.4 \times {10^8}\,$ Joules

C) 6.4 \times {10^9}\,$ Joules

D) 6.4 \times {10^{10}}\,$ Joules

229

MediumJEE Mains2011

Two bodies of masses $m and 4 m are placed at a distance r.$ The gravitational potential at a point on the line joining them where the gravitational field is zero is:

Options:

A) - {{4Gm} \over r}

B) - {{6Gm} \over r}

C) - {{9Gm} \over r}

D) zero

230

MediumJEE Mains2009

The height at which the acceleration due to gravity becomes ${g \over 9} (where g= the acceleration due to gravity on the surface of the earth) in terms of R,$ the radius of the earth, is:

Options:

A) {R \over {\sqrt 2 }}

B) R/2

C) \sqrt 2 \,\,R

D) 2\,R

231

MediumJEE Mains2008

A planet in a distant solar system is $10 times more massive than the earth and its radius is 10 times smaller. Given that the escape velocity from the earth is 11\,\,km\,{s^{ - 1}},$ the escape velocity from the surface of the planet would be

Options:

A) 1.1\,\,km\,{s^{ - 1}}

B) 100\,\,km\,{s^{ - 1}}

C) 110\,\,km\,{s^{ - 1}}

D) 0.11\,\,km\,{s^{ - 1}}

232

MediumJEE Mains2008

This question contains Statement - $1 and Statement - 2. of the four choices given after the statements, choose the one that best describes the two statements. Statement - 1: For a mass M kept at the center of a cube of side 'a', the flux of gravitational field passing through its sides 4\,\pi \,GM. Statement - 2: If the direction of a field due to a point source is radial and its dependence on the distance 'r' from the source is given as {1 \over {{r^2}}},$ its flux through a closed surface depends only on the strength of the source enclosed by the surface and not on the size or shape of the surface.

Options:

A) Statement - $1 is false, Statement - 2$ is true

B) Statement - $1 is true, Statement - 2 is true; Statement - 2 is a correct explanation for Statement - 1

C) Statement - $1 is true, Statement - 2 is true; Statement - 2 is not a correct explanation for Statement - 1

D) Statement - $1 is true, Statement - 2$ is false

233

MediumJEE Mains2007

If ${g_E} and {g_M} are the accelerations due to gravity on the surfaces of the earth and the moon respectively and if Millikan's oil drop experiment could be performed on the two surfaces, one will find the ratio {{electro\,\,ch\arg e\,\,on\,\,the\,\,moon} \over {electronic\,\,ch\arg e\,\,on\,\,the\,\,earth}}\,\,to\,be

Options:

A) {g_M}/{g_E}

B) 1

C) 0

D) {g_E}/{g_M}

234

MediumJEE Mains2005

A particle of mass $10 g is kept on the surface of a uniform sphere of mass 100 kg and radius 10 cm. Find the work to be done against the gravitational force between them to take the particle far away from the sphere (you may take G = 6.67 \times {10^{ - 11}}\,\,N{m^2}/k{g^2}$)

Options:

A) 3.33 \times {10^{ - 10}}\,J

B) 13.34 \times {10^{ - 10}}\,J

C) 6.67 \times {10^{ - 10}}\,J

D) 6.67 \times {10^{ - 9}}\,J

235

MediumJEE Mains2005

The change in the value of $g at a height h above the surface of the earth is the same as at a depth d below the surface of earth. When both d and h$ are much smaller than the radius of earth, then which one of the following is correct?

Options:

A) d = {{3h} \over 2}

B) d = {h \over 2}

C) d = h

D) d = 2\,h

236

MediumJEE Mains2005

Average density of the earth

Options:

A) is a complex function of $g

B) does not depend on $g

C) is inversely proportional to $g

D) is directly proportional to $g

237

MediumJEE Mains2004

A satellite of mass $m revolves around the earth of radius R at a height x from its surface. If g$ is the acceleration due to gravity on the surface of the earth, the orbital speed of the satellite is

Options:

A) {{g{R^2}} \over {R + x}}

B) {{gR} \over {R - x}}

C) {gx}

D) {\left( {{{g{R^2}} \over {R + x}}} \right)^{1/2}}

238

MediumJEE Mains2004

If $g is the acceleration due to gravity on the earth's surface, the gain in the potential energy of an object of mass m raised from the surface of the earth to a height equal to the radius R$ of the earth is

Options:

A) {1 \over 4}mgR

B) 2mgR

C) {1 \over 2}mgR

D) mgR

239

MediumJEE Mains2004

The time period of an earth satellite in circular orbit is independent of

Options:

A) both the mass and radius of the orbit

B) radius of its orbit

C) the mass of the satellite

D) neither the mass of the satellite nor the radius of its orbit

240

MediumJEE Mains2004

Suppose the gravitational force varies inversely as the nth power of distance. Then the time period of a planet in circular orbit of radius $R$ around the sun will be proportional to

Options:

A) {R^n}

B) {R^{\left( {{{n - 1} \over 2}} \right)}}

C) {R^{\left( {{{n + 1} \over 2}} \right)}}

D) {R^{\left( {{{n - 2} \over 2}} \right)}}

241

MediumJEE Mains2003

The time period of satellite of earth is $5 hours. If the separation between the earth and the satellite is increased to 4$ times the previous value, the new time period will become

Options:

A) 10$ hours

B) 80$ hours

C) 40$ hours

D) 20$ hours

242

MediumJEE Mains2003

The escape velocity for a body projected vertically upwards from the surface of earth is $11 km/s. If the body is projected at an angle of {45^ \circ }$ with the vertical, the escape velocity will be

Options:

A) 11\sqrt 2 \,\,km/s

B) 22 km/s

C) 11 km/s

D) {{11} \over {\sqrt 2 }}km/s

243

MediumJEE Mains2003

Two spherical bodies of mass $M and 5M & radii R & 2R respectively are released in free space with initial separation between their centers equal to 12R$. If they attract each other due to gravitational force only, then the distance covered by the smaller body just before collision is

Options:

A) 2.5 R

B) 4.5 R

C) 7.5 R

D) 1.5 R

244

MediumJEE Mains2002

The kinetic energy needed to project a body of mass $m from the earth surface (radius R$) to infinity is

Options:

A) mgR/2

B) 2mgR

C) mgR

D) mgR/4

245

MediumJEE Mains2002

If suddenly the gravitational force of attraction between Earth and a satellite revolving around it becomes zero, then the satellite will

Options:

A) continue to move in its orbit with same velocity

B) move tangentially to the original orbit with the same velocity

C) become stationary in its orbit

D) move towards the earth

246

MediumJEE Mains2002

The escape velocity of a body depends upon mass as

Options:

A) {m^0}

B) {m^1}

C) {m^2}

D) {m^3}

247

MediumJEE Mains2002

Energy required to move a body of mass $m from an orbit of radius 2R to 3R$ is

Options:

A) {{GMm} \over {12{R^2}}}

B) {{GMm} \over {3{R^2}}}

C) {{GMm} \over {8R}}

D) {{GMm} \over {6R}}

248

MediumJEE Mains2025

Three identical spheres of mass m , are placed at the vertices of an equilateral triangle of length a. When released, they interact only through gravitational force and collide after a time \mathrm{T}=4 seconds. If the sides of the triangle are increased to length 2 a and also the masses of the spheres are made 2 m , then they will collide after__________seconds.

Options:

249

EasyJEE Mains2025

A satellite of mass 1000 kg is launched to revolve around the earth in an orbit at a height of 270 km from the earth's surface. Kinetic energy of the satellite in this orbit is____________ \times 10^{10} \mathrm{~J}. (Mass of earth =6 \times 10^{24} \mathrm{~kg}, Radius of earth =6.4 \times 10^6 \mathrm{~m}, Gravitational constant =6.67 \times 10^{-11} \mathrm{Nm}^2 \mathrm{~kg}^{-2} )

Options:

250

MediumJEE Mains2025

Two planets, A and B are orbiting a common star in circular orbits of radii R_A and R_B, respectively, with R_B=2 R_A. The planet B is 4 \sqrt{2} times more massive than planet A. The ratio \left(\frac{\mathrm{L}_{\mathrm{B}}}{\mathrm{L}_{\mathrm{A}}}\right) of angular momentum \left(L_B\right) of planet B to that of planet A\left(L_A\right) is closest to integer ________.

Options:

251

EasyJEE Mains2025

Acceleration due to gravity on the surface of earth is ' g '. If the diameter of earth is reduced to one third of its original value and mass remains unchanged, then the acceleration due to gravity on the surface of the earth is ________ g.

Options:

252

MediumJEE Mains2025

A satellite of mass \frac{M}{2} is revolving around earth in a circular orbit at a height of \frac{R}{3} from earth surface. The angular momentum of the satellite is \mathrm{M} \sqrt{\frac{\mathrm{GMR}}{x}}. The value of x is _________ , where M and R are the mass and radius of earth, respectively. ( G is the gravitational constant)

Options:

253

MediumJEE Mains2024

If the radius of earth is reduced to three-fourth of its present value without change in its mass then value of duration of the day of earth will be ________ hours 30 minutes.

Options:

254

MediumJEE Mains2024

A simple pendulum is placed at a place where its distance from the earth's surface is equal to the radius of the earth. If the length of the string is $4 m, then the time period of small oscillations will be __________ s. [take g=\pi^2 m s^{-2}$]

Options:

255

MediumJEE Mains2023

If the earth suddenly shrinks to $\frac{1}{64}th of its original volume with its mass remaining the same, the period of rotation of earth becomes \frac{24}{x}$h. The value of x is __________.

Options:

256

EasyJEE Mains2022

If the acceleration due to gravity experienced by a point mass at a height h above the surface of earth is same as that of the acceleration due to gravity at a depth $\alpha \mathrm{h}\left(\mathrm{h}<<\mathrm{R}_{\mathrm{e}}\right) from the earth surface. The value of \alpha will be _________. (use \left.\mathrm{R}_{\mathrm{e}}=6400 \mathrm{~km}\right)

Options:

257

EasyJEE Mains2022

Two satellites S1 and S2 are revolving in circular orbits around a planet with radius R1 = 3200 km and R2 = 800 km respectively. The ratio of speed of satellite S1 to be speed of satellite S2 in their respective orbits would be ${1 \over x}$ where x = ___________.

Options:

258

MediumJEE Mains2021

Two satellites revolve around a planet in coplanar circular orbits in anticlockwise direction. Their period of revolutions are 1 hour and 8 hours respectively. The radius of the orbit of nearer satellite is 2 $\times 103 km. The angular speed of the farther satellite as observed from the nearer satellite at the instant when both the satellites are closest is {\pi \over x}rad\,{h^{ - 1}}$ where x is ____________.

Options:

259

MediumJEE Mains2021

A body of mass (2M) splits into four masses (m, M $- m, m, M - m}, which are rearranged to form a square as shown in the figure. The ratio of {M \over m}$ for which, the gravitational potential energy of the system becomes maximum is x : 1. The value of x is ............ .

Options:

260

MediumJEE Mains2021

Suppose two planets (spherical in shape) in radii R and 2R, but mass M and 9M respectively have a centre to centre separation 8 R as shown in the figure. A satellite of mass 'm' is projected from the surface of the planet of mass 'M' directly towards the centre of the second planet. The minimum speed 'v' required for the satellite to reach the surface of the second planet is $\sqrt {{a \over 7}{{GM} \over R}} $ then the value of 'a' is ____________.[Given : The two planets are fixed in their position]

Options:

261

EasyJEE Mains2021

The radius in kilometer to which the present radius of earth (R = 6400 km) to be compressed so that the escape velocity is increased 10 times is ___________.

Options:

262

MediumJEE Mains2021

If one wants to remove all the mass of the earth to infinity in order to break it up completely.The amount of energy that needs to be supplied will be ${x \over 5}{{G{M^2}} \over R}$ where x is __________ (Round off to the Nearest Integer) (M is the mass of earth, R is the radius of earth, G is the gravitational constant)

Options:

263

MediumJEE Mains2021

In the reported figure of earth, the value of acceleration due to gravity is same at point A and C but it is smaller than that of its value at point B (surface of the earth). The value of OA : AB will be x : y. The value of x is ________.

Options:

264

MediumJEE Mains2021

The initial velocity vi required to project a body vertically upward from the surface of the earth to reach a height of 10R, where R is the radius of the earth, may be described in terms of escape velocity ve such that ${v_i} = \sqrt {{x \over y}} \times {v_e}$. The value of x will be ____________.

Options:

265

MediumJEE Mains2020

An asteroid is moving directly towards the centre of the earth. When at a distance of 10R (R is the radius of the earth) from the earths centre, it has a speed of 12 km/s. Neglecting the effect of earths atmosphere, what will be the speed of the asteroid when it hits the surface of the earth (escape velocity from the earth is 11.2 km/s) ? Give your answer to the nearest integer in kilometer/s _____.

Options:

266

MediumJEE Mains2020

A ball is dropped from the top of a 100 m high tower on a planet. In the last ${1 \over 2}s$ before hitting the ground, it covers a distance of 19 m. Acceleration due to gravity (in ms–2) near the surface on that planet is _____.

Options:

267

MediumJEE Mains2016

An astronaut of mass m is working on a satellite orbiting the earth at a distance h from the earth’s surface. The radius of the earth is R, while its mass is M. The gravitational pull FG on the astronaut is :

Options:

A) Zero since astronaut feels weightless

B) 0 < FG < ${{GMm} \over {{R^2}}}

C) {{GMm} \over {{{\left( {R + h} \right)}^2}}} < FG < {{GMm} \over {{R^2}}}

D) FG = ${{GMm} \over {{{\left( {R + h} \right)}^2}}}

268

MediumMHT CET2025

A uniform sphere has radius ' R ' and mass ' M '. The magnitude of gravitational field at distances ' \mathrm{r}_1 ' and ' \mathrm{r}_2 ' from the centre of the sphere are ' E_1 ' and ' E_2 ' respectively. The ratio E_1: E_2 is \left(r_1>R\right. and $\left.r_2

Options:

A) \frac{R^2}{r_1^2 r_2}

B) \frac{R^3}{r_1 r_2}

C) \frac{R^3}{r_1^2 r_2}

D) \frac{R^3}{r_1 r_2^2}

269

MediumMHT CET2025

The depth at which acceleration due to gravity becomes \frac{g^{\prime}}{n} is ( R= radius of earth, \mathrm{g}= acceleration due to gravity) ( \mathrm{n}= integer)

Options:

A) \frac{R(n-1)}{n}

B) \frac{R(n+1)}{n}

C) \frac{R(n-1)^2}{n}

D) \frac{R(n+1)^2}{n}

270

MediumMHT CET2025

A body weighs 45 N on the surface of the earth. The gravitational force on a body due to earth at a height equal to half the radius of earth will be

Options:

A) 20 N

B) 22.5 N

C) 30 N

D) 36 N

271

MediumMHT CET2025

The depth at which the value of acceleration due to gravity becomes \left(\frac{1}{n}\right) times the value at the surface of the earth is ( \mathrm{R}= radius of the earth)

Options:

A) \frac{R(n-1)}{n}

B) \frac{R(n+1)}{n}

C) \frac{\mathrm{Rn}}{(\mathrm{n}-1)}

D) \frac{R}{n}

272

MediumMHT CET2025

Two planets A and B have densities ' \rho_1 ', ' \rho_2 ' and have radii ' r_1 ', ' r_2 ', respectively. The ratio of acceleration due to gravity on A to that of B is

Options:

A) r_1: r_2

B) r_1 \rho_1: r_2 \rho_2

C) \quad r_1^2 \rho_1: r_2^2 \rho_2

D) r_1 \rho_2: r_2 \rho_1

273

MediumMHT CET2025

The gravitational pull of the moon is \left(\frac{1}{6}\right)^{th} of the earth and mass of moon is \left(\frac{1}{8}\right)^{\text {th }} of the earth. This implies that the

Options:

A) radius of moon is (1 / 4)^{\text {th }} of the earth's radius.

B) radius of the earth is (\sqrt{4 / 3})^{\text {th }} of the moon's radius.

C) moon's radius is half that of the earth.

D) radius of the earth is (4 / 3)^{\text {th }} of the moon's radius.

274

MediumMHT CET2025

A uniform solid sphere of mass ' m ' and radius ' r ' is surrounded by a uniform thin spherical shell of radius ' 2 r ' and mass ' m ' then the gravitational field

Options:

A) at a distance of 15 r from the centre is $ \frac{2}{9} \frac{\mathrm{Gm}}{\mathrm{r}^2}

B) at a distance of (2.5)r from the centre is $ \frac{8}{25} \frac{\mathrm{Gm}}{\mathrm{r}^2}

C) at a distance of (1.5)r from the centre is zero.

D) between the sphere and spherical shell is uniform.

275

MediumMHT CET2025

The magnitude of gravitational potential energy of a body at a distance ' R ' from the centre of the earth is ' E '. Its weight at a distance ' 1.5 R ' from the centre of the earth is

Options:

A) \frac{2 \mathrm{E}}{9 \mathrm{R}}

B) \frac{4 E}{5 R}

C) \frac{4 E}{9 R}

D) \frac{2 E}{7 R}

276

MediumMHT CET2025

Time period of a simple pendulum on earth's surface is ' T '. It time period becomes ' xT ' when taken to a height ' 2 R ' above earth's surface. The value of x will be (R= radius of earth)

Options:

A) 2

B) 4

C) 1

D) 3

277

MediumMHT CET2025

Two satellites P and Q go round a planet in circular orbits having radii ' 3 R ' and ' R ' respectively. If the speed of satellite P is ' 2 V ', the speed of the satellite Q will be

Options:

A) 2 \sqrt{3} \mathrm{~V}

B) \frac{2 \mathrm{~V}}{\sqrt{3}}

C) \frac{\mathrm{V}}{2}

D) \frac{\mathrm{V}}{\sqrt{3}}

278

MediumMHT CET2025

The radii of circular orbits of two satellites A and B of the earth are ' 4 R ' and ' R ' respectively, where R is the radius of earth. If the speed of satellite B is 6 V , then the speed of satellite A will be

Options:

A) 3 V

B) 4 V

C) 12 V

D) \frac{3}{4} \mathrm{~V}

279

MediumMHT CET2025

Water rises up to height ' x ' in a capillary tube immersed vertically in water. When the whole arrangement is taken to a depth ' d ' in a mine, the water level rises up to height Y . If R is the radius of the earth then the ratio \mathrm{Y}: \mathrm{x} is

Options:

A) \quad \mathrm{R}:(\mathrm{R}+\mathrm{d})

B) \mathrm{R}:(\mathrm{R}-\mathrm{d})

C) \mathrm{R}:(\mathrm{R}-\mathrm{d})^2

D) \mathrm{R}:(\mathrm{R}+\mathrm{d})^2

280

MediumMHT CET2025

The total energy of a circularly orbiting satellite is

Options:

A) half the kinetic energy of the satellite.

B) half the potential energy of the satellite.

C) twice the kinetic energy of the satellite.

D) twice the potential energy of the satellite.

281

MediumMHT CET2025

The time period of a satellite of earth is 24 hours. If the separation between the earth and the satellite is decreased to one fourth of the previous value then its new time period will become

Options:

A) 3 hours

B) 6 hours

C) 24 hours

D) 12 hours

282

MediumMHT CET2025

A body is projected vertically from earth's surface with \left(\frac{1}{3}\right)^{\mathrm{rd}} of escape velocity. The maximum height reached by the body is ( R= radius of earth)

Options:

A) \frac{\mathrm{R}}{4}

B) \frac{\mathrm{R}}{8}

C) \frac{R}{9}

D) \frac{R}{6}

283

MediumMHT CET2025

The escape velocity of a satellite from the surface of earth does NOT depend on

Options:

A) mass of the earth.

B) mass of the object to be projected.

C) radius of the earth.

D) gravitational constant.

284

MediumMHT CET2025

Two particles of equal mass ' m ' move in a circle of radius ' r ' under the action of their mutual gravitational attraction. The speed of each particle will be ( \mathrm{G}= Universal gravitational constant)

Options:

A) \sqrt{\frac{\mathrm{Gm}}{4 \mathrm{r}}}

B) \sqrt{\frac{\mathrm{Gm}}{\mathrm{r}}}

C) \sqrt{\frac{\mathrm{Gm}}{2\mathrm{r}}}

D) \sqrt{\frac{4 \mathrm{Gm}}{\mathrm{r}}}

285

MediumMHT CET2024

A boy weighs 72 N on the surface of earth. The gravitational force on a body due to earth at a height equal to half the radius of earth will be

Options:

A) 32 N

B) 48 N

C) 96 N

D) 162 N

286

MediumMHT CET2024

A satellite is orbiting just above the surface of the planet of density ' \rho ' with periodic time ' T '. The quantity \mathrm{T}^2 \rho is equal to ( \mathrm{G}= universal gravitational constant)

Options:

A) \frac{4 \pi^2}{G}

B) \frac{3 \pi^2}{G}

C) \frac{3 \pi}{G}

D) \frac{\pi}{G}

287

MediumMHT CET2024

The speed with which the earth would have to rotate about its axis so that a person on the equator would weigh \frac{3}{5} th as much as at present weight is ( \mathrm{g}= gravitational acceleration, \mathrm{R}= equatorial radius of the earth)

Options:

A) \sqrt{\frac{2 g}{5 R}}

B) \sqrt{\frac{3 \mathrm{~g}}{5 \mathrm{R}}}

C) \sqrt{\frac{5 R}{3 g}}

D) \sqrt{\frac{3}{5}} \mathrm{gR}

288

MediumMHT CET2024

A simple pendulum has a periodic time ' \mathrm{T}_1 ' when it is on the surface of earth of radius ' R '. Its periodic time is ' \mathrm{T}_2 ' when it is taken to a height ' R ' above the earth's surface. The value of \frac{T_2}{T_1} is

Options:

A) \sqrt2

B) 1

C) 2

D) \frac{1}{2}

289

MediumMHT CET2024

The minimum energy required to launch a satellite of mass m from the surface of a planet of mass M and radius R in a circular orbit at an altitude of 2 R is

Options:

A) \frac{5 \mathrm{GMm}}{6 \mathrm{R}}

B) \frac{2 \mathrm{GMm}}{3 \mathrm{R}}

C) \frac{\mathrm{GMm}}{2 \mathrm{R}}

D) \frac{\mathrm{GMm}}{3 \mathrm{R}}

290

MediumMHT CET2024

The density of a new planet is twice that of earth. The acceleration due to gravity at the surface of the planet is equal to that at the surface of earth. If R is the radius of earth, then radius of the planet would be

Options:

A) 4 R

B) \mathrm{R} / 2

C) \frac{\mathrm{R}}{4}

D) 2 R

291

MediumMHT CET2024

The weights of an object are measured in a coal mine of depth ' h_1 ', then at sea level of height ' h_2 ' and lastly at the top of a mountain of height ' h_3 ' as W_1, W_2 and W_3 respectively. Which one of the following relation is correct? [h h_1 \ll R, h_3 \gg h_2=R, R= radius of the earth ]

Options:

A) \mathrm{W}_1=\mathrm{W}_2=\mathrm{W}_3

B) \mathrm{W}_1<\mathrm{W}_2<\mathrm{W}_3

C) \mathrm{W}_1>\mathrm{W}_2<\mathrm{W}_3

D) \mathrm{W}_1<\mathrm{W}_2>\mathrm{W}_3

292

MediumMHT CET2024

A satellite of mass ' m ' is revolving around the earth of mass ' M ' in an orbit of radius ' r ' with constant angular velocity ' \omega '. The angular momentum of satellite is ( \mathrm{G}= Universal constant of gravitation)

Options:

A) \mathrm{m}(\mathrm{GMr})^{3 / 2}

B) \mathrm{m}(\mathrm{GMr})

C) \mathrm{m}(\mathrm{GMr})^{1 / 2}

D) \mathrm{m}(\mathrm{GMr})^{-1 / 2}

293

MediumMHT CET2024

For a satellite moving in an orbit around the earth at height ' h ' the ratio of kinetic energy to potential energy is

Options:

A) 2: 1

B) 1: 2

C) 1: \sqrt{2}

D) \sqrt{2}: 1

294

MediumMHT CET2024

A body is projected in vertically upward direction from the surface of the earth of radius ' R ' into space with velocity ' n V_{\mathrm{e}} ' (\mathrm{n}<1). The maximum height from the surface of earth to which a body can reach is

Options:

A) \frac{\mathrm{n}^2 \mathrm{R}}{\left(1-\mathrm{n}^2\right)}

B) \frac{\mathrm{n}^2 \mathrm{R}^2}{(1-\mathrm{n})}

C) \frac{n R^2}{\left(1+n^2\right)}

D) \frac{\mathrm{n}^2 \mathrm{R}^2}{(1+\mathrm{n})}

295

MediumMHT CET2024

The acceleration due to gravity at the surface of the planet is same as that at the surface of the earth, but the density of planet is thrice that of the earth. If 'R' is the radius of the earth, the radius of the planet will be

Options:

A) \frac{\mathrm{R}}{9}

B) \frac{\mathrm{R}}{3}

C) 3R

D) 9R

296

MediumMHT CET2024

The depth 'd' at which the value of acceleration due to gravity becomes \frac{1}{n-1} times the value at the earth's surface is (R= radius of the earth)

Options:

A) R\left(\frac{n}{n-1}\right)

B) R\left(\frac{n-2}{n-1}\right)

C) R\left(\frac{2 n-1}{n}\right)

D) \mathrm{R}\left(\frac{\mathrm{n}-1}{2 \mathrm{n}-1}\right)

297

MediumMHT CET2024

Assuming that the earth is revolving around the sun in circular orbit of radius R , the angular momentum is directly proportional to \mathrm{R}^{\mathrm{n}}. The value of ' n ' is

Options:

A) 2

B) 1.5

C) 1

D) 0.5

298

MediumMHT CET2024

The height ' h ' from the surface of the earth at which the value of ' g ' will be reduced by 64 \% than the value at surface of the earth is ( \mathrm{R}= radius of the earth)

Options:

A) \frac{1}{3} R

B) \frac{2}{3} R

C) \frac{3}{2} R

D) \mathrm{2 R}

299

MediumMHT CET2024

A body starts from rest from a distance \mathrm{R}_0 from the centre of the earth. The velocity acquired by the body when it reaches the surface of the earth will be ( R= radius of earth, M= mass of earth)

Options:

A) 2 \mathrm{GM}\left(\frac{1}{\mathrm{R}}-\frac{1}{\mathrm{R}_0}\right)

B) \sqrt{2 \mathrm{GM}\left(\frac{1}{\mathrm{R}}-\frac{1}{\mathrm{R}_0}\right)}

C) \mathrm{GM}\left(\frac{1}{\mathrm{R}}-\frac{1}{\mathrm{R}_0}\right)

D) 2 \mathrm{GM} \sqrt{\left(\frac{1}{\mathrm{R}}-\frac{1}{\mathrm{R}_0}\right)}

300

MediumMHT CET2024

The radius and mean density of the planet are four times as that of the earth. The ratio of escape velocity at the earth to the escape velocity at a planet is

Options:

A) 1: \sqrt{8}

B) 1: 8

C) 1: \sqrt{3}

D) 1: 3

301

MediumMHT CET2024

A small planet is revolving around a very massive star in a circular orbit of radius ' R ' with a period of revolution ' T '. If the gravitational force between the planet and the star were proportional to 'R^{-5 / 2}', then 'T' would be proportional to

Options:

A) \mathrm{R}^{3 / 2}

B) \mathrm{R}^{3 / 5}

C) \mathrm{R}^{7 / 2}

D) \mathrm{R}^{7 / 4}

302

MediumMHT CET2024

A satellite is revolving around a planet in a circular orbit close to its surface. Let ' \rho ' be the mean density and ' R ' be the radius of the planet. Then the period of the satellite is ( \mathrm{G}= universal constant of gravitation)

Options:

A) \sqrt{\frac{4 \pi}{\rho G}}

B) \sqrt{\frac{\pi}{\rho G}}

C) \sqrt{\frac{3 \pi}{\rho G}}

D) \sqrt{\frac{2 \pi}{\rho G}}

303

MediumMHT CET2024

The radius of the planet is double that of the earth, but their average densities are same. \mathrm{V}_{\mathrm{p}} and V_E are the escape velocities of planet and earth respectively. If \frac{V_P}{V_E}=x, the value of ' x ' is

Options:

A) \frac{1}{4}

B) \frac{1}{2}

C) 2

D) 4

304

MediumMHT CET2024

Two satellites A and B having ratio of masses 3: 1 are revolving in circular orbits of radii ' r ' and ' 4 r '. The ratio of total energy of satellites A to that of B is

Options:

A) 1: 3

B) 3: 1

C) 3: 4

D) 12: 1

305

MediumMHT CET2024

The period of a planet around the sun is 8 times that of earth. The ratio of radius of planet's orbit to the radius of the earth's orbit is

Options:

A) 4

B) 8

C) 16

D) 64

306

MediumMHT CET2024

A pendulum is oscillating with frequency ' n ' on the surface of earth. If it is taken to a depth \frac{R}{4} below the surface of earth, new frequency of oscillation of depth \frac{\mathrm{R}}{4} is ( \mathrm{R}= radius of earth)

Options:

A) \frac{2}{\sqrt{3} n}

B) \frac{\sqrt{3} n}{2}

C) \frac{2 \mathrm{n}}{\sqrt{3}}

D) \frac{\mathrm{n}}{4}

307

MediumMHT CET2024

The escape velocity from earth surface is 11 \mathrm{~km} / \mathrm{s}. The escape velocity from a planet having twice the radius and same mean density as earth is

Options:

A) 22 km/s

B) 11 km/s

C) 5.5 km/s

D) 15.5 km/s

308

MediumMHT CET2024

If ' R ' is the radius of earth \& ' g ' is acceleration due to gravity on earth's surface, then mean density of earth is

Options:

A) \frac{4 \pi G}{3 g R}

B) \frac{3 \pi R}{4 g G}

C) \frac{3 \mathrm{~g}}{4 \pi \mathrm{RG}}

D) \frac{\pi R G}{12 \mathrm{~g}}

309

MediumMHT CET2024

The height ' h ' above the earth's surface at which the value of acceleration due to gravity (\mathrm{g}) becomes \left(\frac{\mathrm{g}}{3}\right) is ( \mathrm{R}= radius of the earth)

Options:

A) (\sqrt{3}+1) R

B) (\sqrt{3}-1) R

C) \sqrt{3} \mathrm{R}

D) 3 \sqrt{R}

310

MediumMHT CET2024

The height at which the weight of the body becomes \frac{1^{\text {th }}}{16} of its weight on the surface of the earth of radius ' R ' is

Options:

A) 2 R

B) 3 R

C) 4 R

D) 5 R

311

MediumMHT CET2024

Two identical metal spheres are kept in contact with each other, each having radius ' R ' cm and ' \rho ' is the density of material of metal spheres. The gravitational force ' F ' of attraction between them is proportional to

Options:

A) \mathrm{R}^3 \rho

B) R^4 \rho^2

C) R^4 \rho

D) \mathrm{R}^3 p^2

312

MediumMHT CET2024

The distance of the two planets A and B from the sun are r_A and r_B respectively. Also r_B is equal to 100 r_A. If the orbital speed of the planet A is ' v ' then the orbital speed of the planet B is

Options:

A) \frac{\mathrm{v}}{10}

B) \frac{v}{2}

C) \sqrt{2} v

D) 10 v

313

MediumMHT CET2024

Earth has mass ' M_1 ' radius ' R_1 ' and for moon mass ' M_2 ' and radius ' R_2 '. Distance between their centres is ' r '. A body of mass ' M ' is placed on the line joining them at a distance \frac{\mathrm{r}}{3} from the centre of the earth. To project a mass ' M ' to escape to infinity, the minimum speed required is

Options:

A) \left[\frac{2 G}{r}\left(M_2+\frac{M_1}{2}\right)\right]^{1 / 2}

B) \left[\frac{4 \mathrm{G}}{\mathrm{r}}\left(\mathrm{M}_1+\frac{\mathrm{M}_2}{2}\right)\right]^{1 / 2}

C) \left[\frac{3 G}{r}\left(M_1+M_2\right)\right]^{1 / 2}

D) \left[\frac{6 G}{r}\left(M_1+\frac{M_2}{2}\right)\right]^{1 / 2}

314

MediumMHT CET2024

The gravitational potential energy required to raise a satellite of mass ' m ' to height ' h ' above the earth's surface is ' \mathrm{E}_1 '. Let the energy required to put this satellite into the orbit at the same height be ' E_2 '. If M and R are the mass and radius of the earth respectively then E_1: E_2 is

Options:

A) \mathrm{h}: \mathrm{R}

B) \mathrm{h}: 2 \mathrm{R}

C) \mathrm{R}: \mathrm{h}

D) 2 \mathrm{~h}: \mathrm{R}

315

MediumMHT CET2024

The height above the earth's surface at which the acceleration due to gravity becomes \left(\frac{1}{n}\right) times the value at the surface is ( R= radius of earth)

Options:

A) \frac{\mathrm{R}}{\sqrt{\mathrm{n}}}

B) \mathrm{R} \cdot \sqrt{\mathrm{n}}

C) \quad(\sqrt{n}+1) R

D) (\sqrt{\mathrm{n}}-1) \mathrm{R}

316

MediumMHT CET2024

The magnitude of gravitational field at distance ' r_1 ' and ' r_2 ' from the centre of a uniform sphere of radius ' R ' and mass ' M ' are ' F_1 ' and ' F_2 ' respectively. The ratio ' \left(F_1 / F_2\right) ' will be (if r_1>R and $r_2

Options:

A) \mathrm{\frac{R^2}{r_1 r_2}}

B) \frac{\mathrm{R}^3}{\mathrm{r}_1 \mathrm{r}_2^2}

C) \frac{\mathrm{R}^3}{\mathrm{r}_1^2 \mathrm{r}_2}

D) \frac{\mathrm{R}^4}{\mathrm{r}_1^2 \mathrm{r}_2^2}

317

MediumMHT CET2023

Earth is assumed to be a sphere of radius R. If '$\mathrm{g}_\phi' is value of effective acceleration due to gravity at latitude 30^{\circ} and 'g' is the value at equator, then the value of \left|g-g_\phi\right| is (\omega is angular velocity of rotation of earth, \cos 30^{\circ}=\frac{\sqrt{3}}{2}$ )

Options:

A) \frac{1}{4} \omega^2 \mathrm{R}

B) \frac{3}{4} \omega^2 \mathrm{R}

C) \omega^2 \mathrm{R}

D) \frac{1}{2} \omega^2 \mathrm{R}

318

MediumMHT CET2023

A body (mass $\mathrm{m} ) starts its motion from rest from a point distant R_0\left(R_0>R\right) from the centre of the earth. The velocity acquired by the body when it reaches the surface of earth will be ( \mathrm{G}= universal constant of gravitation, \mathrm{M}= mass of earth, \mathrm{R}$ = radius of earth)

Options:

A) 2 \mathrm{GM}\left(\frac{1}{\mathrm{R}}-\frac{1}{\mathrm{R}_0}\right)

B) \left[2 \mathrm{GM}\left(\frac{1}{\mathrm{R}}-\frac{1}{\mathrm{R}_0}\right)\right]^{\frac{1}{2}}

C) \mathrm{GM}\left(\frac{1}{\mathrm{R}}-\frac{1}{\mathrm{R}_0}\right)

D) 2 \mathrm{GM}\left[\left(\frac{1}{\mathrm{R}}-\frac{1}{\mathrm{R}_0}\right)\right]^{\frac{1}{2}}

319

MediumMHT CET2023

Considering earth to be a sphere of radius '$R' having uniform density '\rho', then value of acceleration due to gravity 'g' in terms of R, \rho and \mathrm{G}$ is

Options:

A) g=\sqrt{\frac{3 \pi R}{\rho G}}

B) \mathrm{g}=\sqrt{\frac{4}{3} \pi \rho \mathrm{GR}}

C) \mathrm{g}=\frac{4}{3} \pi \rho \mathrm{GR}

D) g=\frac{G M}{\rho R^2}

320

MediumMHT CET2023

The value of acceleration due to gravity at a depth '$d' from the surface of earth and at an altitude 'h$' from the surface of earth are in the ratio

Options:

A) 1: 1

B) \frac{R-2 h}{R-d}

C) \frac{R-d}{R-2 h}

D) \frac{\mathrm{R}-\mathrm{d}}{\mathrm{R}-\mathrm{h}}

321

MediumMHT CET2023

If two planets have their radii in the ratio $x: y and densities in the ratio m: n$, then the acceleration due to gravity on them are in the ratio

Options:

A) \frac{n y}{m x}

B) \frac{m y}{n x}

C) \frac{n x}{m y}

D) \frac{m x}{n y}

322

MediumMHT CET2023

A mine is located at depth $R / 3 below earth's surface. The acceleration due to gravity at that depth in mine is (R= radius of earth, g=$ acceleration due to gravity)

Options:

A) g

B) 3 g

C) \frac{2 g}{3}

D) \frac{g}{3}

323

MediumMHT CET2023

A body of mass '$\mathrm{m}' is raised through a height above the earth's surface so that the increase in potential energy is \frac{\mathrm{mgR}}{5}. The height to which the body is raised is ( \mathrm{R}= radius of earth, \mathrm{g}=$ acceleration due to gravity)

Options:

A) \mathrm{R}

B) \frac{\mathrm{R}}{2}

C) \frac{\mathrm{R}}{4}

D) \frac{\mathrm{R}}{8}

324

MediumMHT CET2023

If two identical spherical bodies of same material and dimensions are kept in contact, the gravitational force between them is proportional to $\mathrm{R}^{\mathrm{X}}, where \mathrm{x} is non zero integer [Given : \mathrm{R}$ is radius of each spherical body]

Options:

A) -$4

B) 4

C) 2

D) -$2

325

MediumMHT CET2023

A body is projected vertically upwards from earth's surface of radius '$R' with velocity equal to \frac{1^{\text {rd }}}{3}$ of escape velocity. The maximum height reached by the body is

Options:

A) \frac{R}{8}

B) \frac{\mathrm{R}}{6}

C) \frac{\mathrm{R}}{4}

D) \frac{\mathrm{R}}{9}

326

MediumMHT CET2023

A simple pendulum is oscillating with frequency '$F' on the surface of the earth. It is taken to a depth \frac{\mathrm{R}}{3} below the surface of earth. ( \mathrm{R}= radius of earth). The frequency of oscillation at depth \mathrm{R} / 3$ is

Options:

A) \frac{2 \mathrm{~F}}{3}

B) \frac{\mathrm{F}}{\sqrt{1.5}}

C) F

D) \frac{\mathrm{F}}{3}

327

MediumMHT CET2023

The depth at which acceleration due to gravity becomes $\frac{\mathrm{g}}{2 \mathrm{n}} is (\mathrm{R}= radius of earth, \mathrm{g}= acceleration due to gravity on earth's surface, \mathrm{n}$ is integer)

Options:

A) \frac{\mathrm{R}(1-2 \mathrm{n})}{\mathrm{n}}

B) \frac{\mathrm{R}(1-\mathrm{n})}{2 \mathrm{n}}

C) \frac{\mathrm{R}(\mathrm{n}-1)}{\mathrm{n}}

D) \frac{\mathrm{R}(2 \mathrm{n}-1)}{2 \mathrm{n}}

328

MediumMHT CET2023

Time period of simple pendulum on earth's surface is '$\mathrm{T}'. Its time period becomes '\mathrm{xT}' when taken to a height \mathrm{R} (equal to earth's radius) above the earth's surface. Then the value of 'x$' will be

Options:

A) 4

B) 2

C) \frac{1}{2}

D) \frac{1}{4}

329

MediumMHT CET2023

The height at which the weight of the body becomes $\left(\frac{1}{9}\right)^{\text {th }} its weight on the surface of earth is (\mathrm{R}=$ radius of earth)

Options:

A) 8R

B) 4R

C) 3R

D) 2R

330

MediumMHT CET2023

Consider a light planet revolving around a massive star in a circular orbit of radius '$r' with time period 'T'. If the gravitational force of attraction between the planet and the star is proportional to \mathrm{r}^{\frac{7}{2}}, then \mathrm{T}^2$ is proportional to

Options:

A) r^{9 / 2}

B) r^{7 / 2}

C) r^{5 / 2}

D) r^{3 / 2}

331

MediumMHT CET2023

The radius of the orbit of a geostationary satellite is (mean radius of earth is '$R', angular velocity about own axis is '\omega' and acceleration due to gravity on earth's surface is 'g$')

Options:

A) \left(\frac{\mathrm{gR}^2}{\omega^2}\right)^{\frac{1}{3}}

B) \left(\frac{\mathrm{gR}^2}{\omega^2}\right)^{\frac{2}{3}}

C) \left(\frac{\mathrm{gR}^2}{\omega^2}\right)^{\frac{1}{2}}

D) \frac{\mathrm{gR}^2}{\omega^2}

332

MediumMHT CET2023

The ratio of energy required to raise a satellite to a height '$h' above the earth's surface to that required to put it into the orbit at the same height is (\mathrm{R}=$ radius of earth)

Options:

A) \frac{2 \mathrm{~h}}{\mathrm{R}}

B) \frac{h}{R}

C) \frac{\mathrm{R}}{\mathrm{h}}

D) \frac{\mathrm{R}}{2 \mathrm{~h}}

333

MediumMHT CET2023

The radius of earth is $6400 \mathrm{~km} and acceleration due to gravity \mathrm{g}=10 \mathrm{~ms}^{-2}. For the weight of body of mass 5 \mathrm{~kg} to be zero on equator, rotational velocity of the earth must be (in \mathrm{rad} / \mathrm{s}$ )

Options:

A) \frac{1}{80}

B) \frac{1}{400}

C) \frac{1}{800}

D) \frac{1}{1600}

334

MediumMHT CET2023

A body of mass '$\mathrm{m}' kg starts falling from a distance 3R above earth's surface. When it reaches a distance 'R' above the surface of the earth of radius 'R' and Mass 'M$', then its kinetic energy is

Options:

A) \frac{2}{3} \frac{\mathrm{GMm}}{\mathrm{R}}

B) \frac{1}{3} \frac{\mathrm{GMm}}{\mathrm{R}}

C) \frac{1}{2} \frac{\mathrm{GMm}}{\mathrm{R}}

D) \frac{1}{4} \frac{\mathrm{GMm}}{\mathrm{R}}

335

MediumMHT CET2023

A body is projected vertically from earth's surface with velocity equal to half the escape velocity. The maximum height reached by the satellite is ( $R$ = radius of earth)

Options:

A) \mathrm{R}

B) \frac{R}{2}

C) \frac{R}{3}

D) \frac{\mathrm{R}}{4}

336

MediumMHT CET2023

A system consists of three particles each of mass '$m_1' placed at the corners of an equilateral triangle of side '\frac{\mathrm{L}}{3}', A particle of mass '\mathrm{m}_2' is placed at the mid point of any one side of the triangle. Due to the system of particles, the force acting on \mathrm{m}_2$ is

Options:

A) \frac{3 \mathrm{Gm}_1 \mathrm{~m}_2}{\mathrm{~L}^2}

B) \frac{6 \mathrm{Gm}_1 \mathrm{~m}_2}{\mathrm{~L}^2}

C) \frac{9 \mathrm{Gm}_1 \mathrm{~m}_2}{\mathrm{~L}^2}

D) \frac{12 \mathrm{Gm}_1 \mathrm{~m}_2}{\mathrm{~L}^2}

337

MediumMHT CET2023

A satellite moves in a stable circular orbit round the earth if (where $\mathrm{V}_{\mathrm{H}}, \mathrm{V}_{\mathrm{c}} and \mathrm{V}_{\mathrm{e}}$ are the horizontal velocity, critical velocity and escape velocity respectively)

Options:

A) \mathrm{V}_{\mathrm{H}}<\mathrm{V}_{\mathrm{c}}

B) \mathrm{V}_{\mathrm{H}}=\mathrm{V}_{\mathrm{e}}

C) V_H=V_c

D) V_H>V_e

338

MediumMHT CET2023

There is a second's pendulum on the surface of earth. It is taken to the surface of planet whose mass and radius are twice that of earth. The period of oscillation of second's pendulum on the planet will be

Options:

A) 2 \sqrt{2} \mathrm{~s}

B) 2 \mathrm{~s}

C) \frac{1}{\sqrt{2}} \mathrm{~s}

D) \frac{1}{2} \mathrm{~s}

339

MediumMHT CET2023

For a satellite orbiting around the earth in a circular orbit, the ratio of potential energy to kinetic energy at same height is

Options:

A) \frac{1}{\sqrt{2}}

B) \frac{1}{2}

C) \sqrt{2}

D) 2

340

MediumMHT CET2023

Periodic time of a satellite revolving above the earth's surface at a height equal to radius of the earth '$R' is [ g=$ acceleration due to gravity]

Options:

A) 2 \pi \sqrt{\frac{2 R}{g}}

B) 4 \pi \sqrt{\frac{2 R}{g}}

C) 2 \pi \sqrt{\frac{R}{g}}

D) 8 \pi \sqrt{\frac{\mathrm{R}}{\mathrm{g}}}

341

MediumMHT CET2023

Consider a planet whose density is same as that of the earth but whose radius is three times the radius '$R' of the earth. The acceleration due to gravity '\mathrm{g}_{\mathrm{n}}' on the surface of planet is \mathrm{g}_{\mathrm{n}}=\mathrm{x}. \mathrm{g} where \mathrm{g} is acceleration due to gravity on surface of earth. The value of '\mathrm{x}$' is

Options:

A) 9

B) 3

C) \frac{1}{3}

D) \frac{1}{9}

342

MediumMHT CET2023

A thin rod of length '$L' is bent in the form of a circle. Its mass is 'M'. What force will act on mass 'm' placed at the centre of this circle? ( \mathrm{G}=$ constant of gravitation)

Options:

A) zero

B) \frac{\mathrm{GMm}}{4 \mathrm{~L}^2 \pi^2}

C) \frac{4 \pi^2 \mathrm{GMm}}{\mathrm{L}}

D) \frac{2 \mathrm{GMm}}{\mathrm{L}^2}

343

MediumMHT CET2023

A body weighs $300 \mathrm{~N} on the surface of the earth. How much will it weigh at a distance \frac{R}{2} below the surface of earth? ( R \rightarrow$ Radius of earth)

Options:

A) 300 \mathrm{~N}

B) 250 \mathrm{~N}

C) 200 \mathrm{~N}

D) 150 \mathrm{~N}

344

MediumMHT CET2023

A seconds pendulum is placed in a space laboratory orbiting round the earth at a height '$3 \mathrm{R}' from the earth's surface. The time period of the pendulum will be ( R=$ radius of earth)

Options:

A) zero

B) \frac{2}{3} \mathrm{~s}

C) 4 \mathrm{~s}

D) infinite

345

MediumMHT CET2022

A body weighs $500 \mathrm{~N} on the surface of the earth. At what distance below the surface of the earth it weighs 250 \mathrm{~N} ? (Radius of earth, \mathrm{R}=6400 \mathrm{~km}$ )

Options:

A) 6400 km

B) 800 km

C) 1600 km

D) 3200 km

346

MediumMHT CET2022

The masses and radii of the moon and the earth are $\mathrm{M_1, R_1} and \mathrm{M_2, R_2} respectively. Their centres are at a distance \mathrm{d} apart. What should be the minimum speed with which a body of mass 'm$' should be projected from a point midway between their centres, so as to escape to infinity?

Options:

A) \frac{\mathrm{G}\left(\mathrm{M}_1+\mathrm{M}_2\right)}{\mathrm{d}}

B) \sqrt[2]{\frac{G\left(M_1+M_2\right)}{d}}

C) \sqrt{\frac{G d}{M_1+M_2}}

D) \sqrt{\frac{M_1+M_2}{G d}}

347

MediumMHT CET2021

The average density of the earth is [g is acceleration due to gravity]

Options:

A) inversely proportional to g$^2$.

B) directly proportional to $\mathrm{g}$.

C) inversely proportional to $\mathrm{g}$.

D) directly proportional to $\mathrm{g}^2$.

348

MediumMHT CET2021

The depth from the surface of the earth of radius $\mathrm{R}, at which acceleration due to gravity will be 60 \%$ of the value on the earth surface is

Options:

A) \frac{2 R}{3}

B) \frac{2 R}{5}

C) \frac{3 R}{5}

D) \frac{5 R}{3}

349

MediumMHT CET2021

Three point masses, each of mass 'm' are kept at the corners of an equilateral triangle of side 'L'. The system rotates about the centre of the triangle without any change in the separation of masses during rotation. The period of rotation is directly proportional to $\left(\cos 30^{\circ}=\frac{\sqrt{3}}{2}\right)

Options:

A) L

B) \mathrm{L}^{1 / 2}

C) \mathrm{L}^{3 / 2}

D) \mathrm{L}^{-2}

350

MediumMHT CET2021

The length of the seconds pendulum is lm on earth. If the mass and diameter of the planet is 1.5 times that of the earth, the length of the seconds pendulum on the planet will be nearly

Options:

A) 0.67 m

B) 0.45 m

C) 0.60 m

D) 0.76 m

351

MediumMHT CET2021

If the horizontal velocity given to a satellite is greater than critical velocity but less than the escape velocity at the height, then the satellite will

Options:

A) be lost in space

B) falls on the earth along parabolic path

C) revolve in circular orbit

D) revolve in elliptical orbit

352

MediumMHT CET2021

The period of revolution of planet $\mathrm{A} around the sun is 8 times that of planet \mathrm{B}$. How many times the distance of A from the sun is greater than that of B from the sun?

Options:

A) 5 times

B) 2 times

C) 3 times

D) 6 times

353

MediumMHT CET2021

The time period of a satellite of earth is 5 hours. If the separation between the earth and the satellite will satellite is increased to four times the previous value, the new time period of the satellite will be

Options:

A) 20 hours

B) 40 hours

C) 80 hours

D) 10 hours

354

MediumMHT CET2021

A body of mass 'M' and radius 'R', situated on the surface of the earth becomes weightless at its equator when the rotational kinetic energy of the earth reaches a critical value 'K'. The value of 'K' is given by [Assume the earth as a solid sphere, g = gravitational acceleration on the earth's surfacde]

Options:

A) \frac{1}{2} \mathrm{MgR}

B) \frac{1}{3} \mathrm{MgR}

C) \frac{1}{4} \mathrm{MgR}

D) \frac{1}{5} \mathrm{MgR}

355

MediumMHT CET2021

The mass of a spherical planet is 4 times the mass of the earth, but its radius (R) is same as that of the earth. How much work is done is lifting a body of mass 5 kg through a distance of 2 m on the planet ? (g = 10 ms$^{-2}$)

Options:

A) 400 J

B) 200 J

C) 800 J

D) 300 J

356

MediumMHT CET2021

The radius of a planet is twice the radius of the earth. Both have almost equal average mass densities. If '$V_P' and 'V_E$' are escape velocities of the planet and the earth respectively, then

Options:

A) \mathrm{V}_{\mathrm{E}}=1.5 \mathrm{~V}_{\mathrm{P}}

B) \mathrm{V}_{\mathrm{P}}=1.5 \mathrm{~V}_{\mathrm{E}}

C) \mathrm{V_P=2 V_E}

D) \mathrm{V_E=3 V_P}

357

MediumMHT CET2021

Two satellites of same mass are launched in circular orbits at heights '$R' and '2 R' above the surface of the earth. The ratio of their kinetic energies is (R=$ radius of the earth)

Options:

A) 1: 3

B) 3: 2

C) 4: 9

D) 9: 4

358

MediumMHT CET2021

At a height 'R' above the earth's surface the gravitational acceleration is (R = radius of earth, g = acceleration due to gravity on earth's surface)

Options:

A) g

B) \frac{\mathrm{g}}{8}

C) \frac{g}{4}

D) \frac{g}{2}

359

MediumMHT CET2021

The mass of a planet is six times that of the earth. The radius of the planet is twice that of the earth. If the escape velocity from the earth is '$V_e$', then the escape velocity from the planet is

Options:

A) \sqrt{3} \mathrm{~V}_{\mathrm{e}}

B) \sqrt{2} \mathrm{~V}_{\mathrm{e}}

C) \mathrm{V}_{\mathrm{e}}

D) \sqrt{5} \mathrm{~V}_{\mathrm{e}}

360

MediumMHT CET2021

For a body of mass '$m', the acceleration due to gravity at a distance 'R' from the surface of the earth is \left(\frac{g}{4}\right). Its value at a distance \left(\frac{R}{2}\right) from the surface of the earth is ( R= radius of the earth, g=$ acceleration due to gravity)

Options:

A) \left(\frac{g}{8}\right)

B) \left(\frac{9 g}{4}\right)

C) \left(\frac{4 g}{9}\right)

D) \left(\frac{\mathrm{g}}{2}\right)

361

MediumMHT CET2021

The ratio of energy required to raise a satellite of mass '$m' to height 'h' above the earth's surface to that required to put it into the orbit at same height is [ \mathrm{R}=$ radius of earth]

Options:

A) \frac{h}{R}

B) \frac{2 h}{\mathrm{R}^2}

C) \frac{3 \mathrm{~h}}{\mathrm{R}^2}

D) \frac{2 \mathrm{~h}}{\mathrm{R}}

362

MediumMHT CET2021

A pendulum is oscillating with frequency '$n' on the surface of the earth. It is taken to a depth \frac{R}{2} below the surface of earth. New frequency of oscillation at depth \frac{R}{2} is [ R$ is the radius of earth]

Options:

A) \mathrm{\frac{n}{3}}

B) \frac{\mathrm{n}}{\sqrt{2}}

C) \mathrm{2 n}

D) \frac{\mathrm{n}}{2}

363

MediumMHT CET2021

When the value of acceleration due to gravity '$g' becomes \frac{g}{3} above surface of height 'h' then relation between 'h' and 'R' is ( \mathrm{R}=$ radius of earth)

Options:

A) \mathrm{h}=\frac{\mathrm{R}}{\sqrt{3}-1}

B) \mathrm{h}=\frac{\sqrt{3}}{\mathrm{R}}

C) h=(\sqrt{2}-1) R

D) \mathrm{h}=(\sqrt{3}-1) \mathrm{R}

364

MediumMHT CET2021

A particle of mass '$m' is kept at rest at a height 3 R from the surface of earth, where 'R' is radius of earth and 'M' is the mass of earth. The minimum speed with which it should be projected, so that it does not return back is ( g=$ acceleration due to gravity on the earth's surface)

Options:

A) \left[\frac{\mathrm{GM}}{2 \mathrm{R}}\right]^{1 / 2}

B) \left[\frac{\mathrm{gR}}{4}\right]^{1 / 2}

C) \left[\frac{2 \mathrm{~g}}{\mathrm{R}}\right]^{1 / 2}

D) \left[\frac{\mathrm{GM}}{\mathrm{R}}\right]^{1 / 2}

365

MediumMHT CET2021

A body is projected from earth's surface with thrice the escape velocity from the surface of the earth. What will be its velocity when it will escape the gravitational pull?

Options:

A) 2 \mathrm{~V}_{\mathrm{e}}

B) 4 \mathrm{~V}_{\mathrm{e}}

C) 2 \sqrt{2} \mathrm{~V}_{\mathrm{e}}

D) \frac{\mathrm{V}_{\mathrm{e}}}{2}

366

MediumMHT CET2021

The depth at which acceleration due to gravity becomes $\frac{\mathrm{g}}{\mathrm{n}} is [ \mathrm{R} = radius of earth, \mathrm{g}= acceleration due to gravity, \mathrm{n}= integer ]

Options:

A) \frac{\mathrm{R}(\mathrm{n}-1)}{\mathrm{n}}

B) \frac{(\mathrm{n}-1)}{\mathrm{nR}}

C) \frac{\mathrm{Rn}}{(\mathrm{n}-1)}

D) \frac{\mathrm{n}}{\mathrm{R}(\mathrm{n}-1)}

367

MediumMHT CET2021

The depth 'd' below the surface of the earth where the value of acceleration due to gravity becomes $\left(\frac{1}{n}\right) times the value at the surface of the earth is (R=$ radius of the earth)

Options:

A) \mathrm{R}\left(\frac{\mathrm{n}-1}{\mathrm{n}}\right)

B) R\left(\frac{n}{n+1}\right)

C) \frac{R}{n}

D) \frac{\mathrm{R}}{\mathrm{n}^2}

368

MediumMHT CET2021

The orbital velocity of an artificial satellite in a circular orbit just above the earth's surface is 'V'. For the satellite orbiting at an altitude of half the earth's radius, the orbital velocity is

Options:

A) \frac{3}{2}$V

B) \sqrt{\frac{3}{2}}$V

C) \sqrt{\frac{2}{3}}$V

D) \frac{2}{3}$V

369

MediumMHT CET2020

Earth has mass M_1 and radius R_1. Moon has mass M_2 and radius R_2. Distance between their centre is r. A body of mass M is placed on the line joining them at a distance \frac{r}{3} from centre of the earth. To project the mass M to escape to infinity, the minimum speed required is

Options:

A) \left[\frac{3 G}{r}\left(M_1+\frac{M_2}{2}\right)\right]^{\frac{1}{2}}

B) \left[\frac{6 G}{r}\left(M_1+\frac{M_2}{2}\right)\right]^{\frac{1}{2}}

C) \left[\frac{6 G}{r}\left(M_1-\frac{M_2}{2}\right)\right]^{\frac{1}{2}}

D) \left[\frac{3 G}{r}\left(M_1-\frac{M_2}{2}\right)\right]^{\frac{1}{2}}

370

MediumMHT CET2020

The escape velocity of a body from any planet, whose mass is six times the mass of earth and radius is twice the radius of earth will (v_e = escape velocity of a body from the earth's surface)

Options:

A) 2 \sqrt{2} v_e

B) \frac{3}{2} v_e

C) 2 v_e

D) \sqrt{3} v_e

371

MediumMHT CET2020

The ratio of energy required to raise a satellite of mass $m to a height h above the earth's surface of that required to put it into the orbit at same height is [R=$ radius of the earth]

Options:

A) \frac{h}{R}

B) \frac{3 h}{R}

C) \frac{4 h}{R}

D) \frac{2 h}{R}

372

MediumMHT CET2020

As we go from the equator of the earth to pole of the earth, the value of acceleration due to gravity

Options:

A) decreases

B) decreases up to latitude of $45^{\circ}$ and increases thereafter

C) remains same

D) increases

373

MediumMHT CET2020

The mass of earth is 81 times the mass of the moon and the distance between their centres is $R$. The distance from the centre of the earth, where gravitational force will be zero is

Options:

A) \frac{R}{4}

B) \frac{R}{2}

C) \frac{9 R}{10}

D) \frac{R}{81}

374

MediumMHT CET2020

A body is thrown from the surface of the earth velocity $\mathrm{v} / \mathrm{s}. The maximum height above the earth's surface upto which it will reach is (R= radius of earth, g=$ acceleration due to gravity)

Options:

A) \frac{v R}{2 g R-v}

B) \frac{2 g A}{v^2(R-1)}

C) \frac{v R^2}{g R-v}

D) \frac{v^2 R}{2 g R-v^2}

375

MediumMHT CET2019

Consider a particle of mass m suspended by a string at the equator. Let R and M denote radius and mass of the earth. If \omega is the angular velocity of rotation of the earth about its own axis, then the tension on the string will be \left(\cos 0^{\circ}=1\right)

Options:

A) \frac{G M m}{R^2}

B) \frac{G M m}{2 R^2}

C) \frac{G M m}{2 R^2}+m \omega^2 R

D) \frac{G M m}{R^2}-m \omega^2 R

376

MediumMHT CET2019

A hole is drilled half way to the centre of the earth. A body weighs 300 N on the surface of the earth. How much will, it weigh at the bottom of the hole?

Options:

A) 200 N

B) 250 N

C) 120 N

D) 150 N

377

MediumMHT CET2019

What is the minimum energy required to launch a satellite of mass ' m ' from the surface of the earth of mass ' M ' and radius ' R ' at an altitude 2 R ?

Options:

A) \frac{G M m}{2 R}

B) \frac{2 G M m}{3 R}

C) \frac{G M m}{3 R}

D) \frac{5 G M m}{6 R}

378

MediumMHT CET2019

The radius of the earth and the radius of orbit around the sun are 6371 km and 149 \times 10^6 \mathrm{~km} respectively. The order of magnitude of the diameter of the orbit is greater than that of earth by

Options:

A) 10^3

B) 10^2

C) 10^4

D) 10^5

379

MediumMHT CET2019

If W_1, W_2 and W_3 represent the work done in moving a particle from A to B along three different paths 1,2 and 3 (as shown in fig) in the gravitational field of the point mass ' m '. Find the correct relation between ' W_1 ', ' W_2 ' and ' W_3 '

Options:

A) W_1< W_3< W_2

B) W_1< W_2< W_3

C) W_1=W_2=W_3

D) W_1>W_3>W_2

380

MediumMHT CET2019

The kinetic energy of a revolving satellite (mass m ) at a height equal to thrice the radius of the earth (R) is

Options:

A) \frac{m g R}{8}

B) \frac{m g R}{16}

C) \frac{m g R}{2}

D) \frac{m g R}{4}

381

MediumMHT CET2019

A body is projected vertically from the surface of the earth of radius ' R ' with velocity equal to half of the escape velocity. The maximum height reached by the body is

Options:

A) \frac{R}{5}

B) \frac{R}{3}

C) \frac{R}{2}

D) \frac{R}{4}

382

EasyNEET2025

The radius of Martian orbit around the Sun is about 4 times the radius of the orbit of Mercury. The Martian year is 687 Earth days. Then which of the following is the length of 1 year on Mercury?

Options:

A) 172 earth days

B) 124 earth days

C) 88 earth days

D) 225 earth days

383

MediumNEET2025

A body weighs 48 N on the surface of the earth. The gravitational force experienced by the body due to the earth at a height equal to one-third the radius of the earth from its surface is:

Options:

A) 32 N

B) 36 N

C) 16 N

D) 27 N

384

MediumNEET2024

The escape velocity for earth is $v$. A planet having 9 times mass that of earth and radius, 16 times that of earth, has the escape velocity of:

Options:

A) \frac{v}{3}

B) \frac{2 v}{3}

C) \frac{3 v}{4}

D) \frac{9 v}{4}

385

MediumNEET2024

An object of mass $100 \mathrm{~kg} falls from point A to B as shown in figure. The change in its weight, corrected to the nearest integer is (R_E$ is the radius of the earth)

Options:

A) 49 N

B) 89 N

C) 5 N

D) 10 N

386

MediumNEET2024

The mass of a planet is $\frac{1}{10}$th that of the earth and its diameter is half that of the earth. The acceleration due to gravity on that planet is:

Options:

A) 19.6 \mathrm{~m} \mathrm{~s}^{-2}

B) 9.8 \mathrm{~m} \mathrm{~s}^{-2}

C) 4.9 \mathrm{~m} \mathrm{~s}^{-2}

D) 3.92 \mathrm{~m} \mathrm{~s}^{-2}

387

MediumNEET2024

The minimum energy required to launch a satellite of mass $m from the surface of earth of mass M and radius R in a circular orbit at an altitude of 2 R$ from the surface of the earth is:

Options:

A) \frac{5 G m M}{6 R}

B) \frac{2 G m M}{3 R}

C) \frac{G m M}{2 R}

D) \frac{G m M}{3 R}

388

MediumNEET2023

The escape velocity of a body on the earth surface is $11.2 \mathrm{~km} / \mathrm{s}. If the same body is projected upward with velocity 22.4 \mathrm{~km} / \mathrm{s}$, the velocity of this body at infinite distance from the centre of the earth will be:

Options:

A) 11.2 \sqrt{2} \mathrm{~km} / \mathrm{s}

B) Zero

C) 11.2 \mathrm{~km} / \mathrm{s}

D) 11.2 \sqrt{3} \mathrm{~km} / \mathrm{s}

389

MediumNEET2023

If $\mathrm{R} is the radius of the earth and \mathrm{g}$ is the acceleration due to gravity on the earth surface. Then the mean density of the earth will be :

Options:

A) \frac{\pi \mathrm{RG}}{12 \mathrm{~g}}

B) \frac{3 \pi R}{4 g G}

C) \frac{3 g}{4 \pi R G}

D) \frac{4 \pi \mathrm{G}}{3 g R}

390

MediumNEET2023

Two bodies of mass $m and 9 m are placed at a distance R. The gravitational potential on the line joining the bodies where the gravitational field equals zero, will be (G=$ gravitational constant) :

Options:

A) -\frac{12 G m}{R}

B) -\frac{16 G m}{R}

C) -\frac{20 G m}{R}

D) -\frac{8 G m}{R}

391

MediumNEET2023

A satellite is orbiting just above the surface of the earth with period $T. If d is the density of the earth and G is the universal constant of gravitation, the quantity \frac{3 \pi}{G d}$ represents :

Options:

A) T^{2}

B) T^{3}

C) \sqrt{T}

D) T

392

MediumNEET2022

A gravitational field is present in a region and a mass is shifted from A to B through different paths as shown. If W1, W2 and W3 represent the work done by the gravitational force along the respective paths, then :

Options:

A) W1 < W2 < W3

B) W1 = W2 = W3

C) W1 > W2 > W3

D) W1 > W3 > W2

393

MediumNEET2022

In a gravitational field, the gravitational potential is given by, $V = - {K \over x}$ (J/Kg). The gravitational field intensity at point (2, 0, 3) m is

Options:

A) + {K \over 4}

B) + {K \over 2}

C) - {K \over 2}

D) - {K \over 4}

394

MediumNEET2022

A body of mass 60 g experiences a gravitational force of 3.0 N, when placed at a particular point. The magnitude of the gravitational field intensity at that point is

Options:

A) 0.05 N/kg

B) 50 N/kg

C) 20 N/kg

D) 180 N/kg

395

MediumNEET2022

Match List - I with List - II List - I List - II (a) Gravitational constant (G) (i) $[{L^2}{T^{ - 2}}] (b) Gravitational potential energy (ii) [{M^{ - 1}}{L^3}{T^{ - 2}}] (c) Gravitational potential (iii) [L{T^{ - 2}}] (d) Gravitational intensity (iv) [M{L^2}{T^{ - 2}}]$ Choose the correct answer from the options given below

Options:

A) (a)-(ii), (b)-(i), (c)-(iv), (d)-(iii)

B) (a)-(ii), (b)-(iv), (c)-(i), (d)-(iii)

C) (a)-(ii), (b)-(iv), (c)-(iii), (d)-(i)

D) (a)-(iv), (b)-(ii), (c)-(i), (d)-(iii)

396

MediumNEET2021

The escape velocity from the Earth's surface is v. The escape velocity from the surface of another planet having a radius, four times that of Earth and same mass density is :

Options:

A) 4$\upsilon

B) \upsilon

C) 2$\upsilon

D) 3$\upsilon

397

MediumNEET2021

A particle of mass 'm' is projected with a velocity $\upsilon $ = kVe = (k < 1) from the surface of the earth. (Ve = escape velocity)The maximum height above the surface reached by the particle is :

Options:

A) {{R{k^2}} \over {1 - {k^2}}}

B) R{\left( {{k \over {1 - {k^2}}}} \right)^2}

C) R{\left( {{k \over {1 + {k^2}}}} \right)^2}

D) {{{R^2}k} \over {1 + k}}

398

MediumNEET2020

A body weighs 72 N on the surface of the earth. What is the gravitational force on it, at a height equal to half the radius of the earth?

Options:

A) 32 N

B) 30 N

C) 24 N

D) 48 N

399

MediumNEET2019

The work done to raise a mass m from the surface of the earth to a height h, which is equal to the radius of the earth, is :

Options:

A) mgR

B) 2mgR

C) {1 \over 2}$mgR

D) {3 \over 2}$mgR

400

MediumNEET2019

A body weight 200 N on the surface of earth. How much will it weight half way down to the centre of the earth?

Options:

A) 200 N

B) 250 N

C) 100 N

D) 150 N

401

MediumNEET2018

If the mass of the Sun were ten times smaller and the universal gravitational constant were ten times larger in magnitude, which of the following is not correct?

Options:

A) Raindrops will fall faster.

B) Walking on the ground would become more difficult.

C) Time period of a simple pendulum on the Earth would decrease.

D) g on the Earth will not change.

402

MediumNEET2018

The kinetic energies of a planet in an elliptical orbit about the Sun, at positions A, B and C are KA, KB and KC, respectively. AC is the major axis and SB is perpendicular to AC at the position of the Sun S as shown in the figure. Then

Options:

A) KA < KB < KC

B) KA > KB > KC

C) KB < KA < KC

D) KB > KA > KC

403

MediumNEET2017

The acceleration due to gravity at a height at a height 1 km above the rearth is the same as at a depth d below the surface of earth. Then

Options:

A) d = 1 km

B) d = ${3 \over 2}$ km

C) d = 2 km

D) d = ${1 \over 2}$ km

404

MediumNEET2017

Two astronauts are floating in gravitational free space after having lost contact with their spaceship. The two will

Options:

A) move towards each other.

B) move away from each other.

C) will become stationary.

D) keep floating at the same distance between them.

405

MediumNEET2016

A satellite of mass m is orbiting the earth (of radius R) at a height h from its surface. The total energy of the satellite in terms of g0, the value of acceleration due to gravity at the earth's surface, is

Options:

A) {{m{g_0}{R^2}} \over {2\left( {R + h} \right)}}

B) - {{m{g_0}{R^2}} \over {2\left( {R + h} \right)}}

C) {{2m{g_0}{R^2}} \over {R + h}}

D) - {{2m{g_0}{R^2}} \over {R + h}}

406

MediumNEET2016

Starting from the centre of the earth having radius R, the variation of g (acceleration due to gravity) is shown by

Options:

407

MediumNEET2016

The ratio of escape velocity at earth (ve) to the escape velocity at a planet (vp) whose radius and mean density are twice as that of earth is

Options:

A) 1 : 4

B) 1 : $\sqrt 2

C) 1 : 2

D) 1 : 2$\sqrt 2

408

MediumNEET2016

At what height from the surface of earth the gravitation potential and the value of g are $-5.4 \times 107 J kg-1 and 6.0 m s-$2 respectively? Take the radius of earth as 6400 km.

Options:

A) 1400 km

B) 2000 km

C) 2600 km

D) 1600 km

409

MediumNEET2015

A satellite S is moving in an elliptical orbit around the earth. The mass of the satellite is very small compared to the mass of the earth. Then,

Options:

A) the linear momentum of S remains constant in magnitude.

B) the acceleration of S is always directed towards the centre of the earth.

C) the angular momentum of S about the centre of the earth changes in direction, but its magnitude remains constant.

D) the total mechanical energy of S varies periodically with time.

410

MediumNEET2015

A remote-sensing satellite of earth revolves in a circular orbit at a height of 0.25 $ \times 106 m above the surface of earth. If earth's radius is 6.38 \times 106 m and g = 9.8 ms-$2, then the orbital speed of the satellite is

Options:

A) 9.13 km s$-$1

B) 6.67 km s$-$1

C) 7.76 km s$-$1

D) 8.56 km s$-$1

411

MediumNEET2015

Kepler's third law states that square of period of revoluation (T) of a planet around the sun, is proportional to third power of average distance r between sun and planet i.e. T2 = Kr3 here K is constant. If the masses of sun and planet are M and m respectively then as per Newton's law of gravitation force of attraction between them is F = ${{GMm} \over {{r^2}}}$, here G is gravitational constant. The relation between G and K is described as

Options:

A) K = G

B) K = ${1 \over G}

C) GK = 4$\pi $2

D) GMK = 4$\pi $2

412

MediumNEET2014

Dependence of intensity of gravitational field (E) of earth with distance (r) from centre of earth is correctly represented by

Options:

413

MediumNEET2014

A black hole is an object whose gravitational field is so strong that even light cannot escape from it. To what approximate radius would earth (mass = 5.98 $ \times $ 1024 kg) have to be compressed to be a black hole?

Options:

A) 10$-$9 m

B) 10$-$6 m

C) 10$-$2 m

D) 100 m

414

MediumNEET2013

The radius of a planet is twice the radius of earth. Both have almost equal average mass densities. VP and VE are escape velocities of the planet and the earth, respectively, then

Options:

A) VP = 1.5 VE

B) VP = 2 VE

C) VE = 3 VP

D) VE = 1.5 VP

415

MediumNEET2013

A particle of mass 'm' is kept at rest at a height 3R from the surface of earth, where 'R' is radius of earth and 'M' is mass of earth. The minimum speed with which it should be projected , so that it does not return back, is (g is acceleration due to gravity on the surface of earth)

Options:

A) {\left( {{{GM} \over {2R}}} \right)^{1/2}}

B) {\left( {{{gR} \over 4}} \right)^{1/2}}

C) {\left( {{{2g} \over R}} \right)^{1/2}}

D) {\left( {{{GM} \over R}} \right)^{1/2}}

416

MediumNEET2013

Infinite number of bodies, each of mass 2 kg are situated on x-axis at distance 1 m, 2 m, 4 m, 8 m, . . . , respectively, from the origin. The resulting gravitational potential due to this system at the origin will be

Options:

A) - {4 \over 3}$ G

B) -$ 4G

C) -$ G

D) - {8 \over 3}$ G

417

MediumNEET2013

A body of mass 'm' is taken from the earth's surface to the height equal to twice the radius (R) of the earth. The change in potential energy of body will be

Options:

A) 3mgR

B) {1 \over 3}$mgR

C) mg2R

D) {2 \over 3}$ mgR

418

MediumNEET2012

If ${v_e} is escape velocity and {v_o}$ is orbital velocity of a satellite for orbit close to the earth's surface, then these are related by

Options:

A) {v_o} = \sqrt {2{v_e}}

B) vo $=$ ve

C) {v_e} = \sqrt {2{v_o}}

D) {v_e} = \sqrt 2 {v_o}

419

MediumNEET2012

Which one of the following plots represents the variation of gravitiational field on a particle with distance r due to a thin spherical shell of radius R? (r is measured from the centre of the spherical shell)

Options:

420

MediumNEET2012

A spherical planet has a mass MP and diameter DP. A particle of mass m falling freely near the surface of this planet will experience an acceleration due to gravity, equal to

Options:

A) {{4G{M_P}} \over {D_P^2}}

B) {{G{M_P}m} \over {D_P^2}}

C) {{G{M_P}} \over {D_P^2}}

D) {{4G{M_P}m} \over {D_P^2}}

421

MediumNEET2012

The height at which the weight of a body becomes ${\left( {{1 \over {16}}} \right)^{th}}$, its weight on the surface of earth (radius R), is

Options:

A) 5R

B) 15R

C) 3R

D) 4R

422

MediumNEET2012

A geostationary satellite is orbiting the earth at a height of 5R above the surface of the earth, R being the radius of the earth. The time period of another satellite in hours at a height of 2R from the surface of the earth is

Options:

A) 5

B) 10

C) 6$\sqrt 2

D) {6 \over {\sqrt 2 }}

423

MediumNEET2011

A particle of mass M is situated at the centre of a spherical shell of same mass and radius a. The magnitude of the gravitational potential at a point sutuated at a/2 distance from the centre, will be :

Options:

A) {{GM} \over a}

B) {{2GM} \over a}

C) {{3GM} \over a}

D) {{4GM} \over a}

424

MediumNEET2011

A particle of mass m is thrown upwards from the surface of the earth, with a velocity u. The mass and the radius of the earth are, respectively, M and R. G is gravitational constant and g is acceleration due to gravity on the surface of the earth. The minimum value of u so that the particle does not return back to earth, is

Options:

A) \sqrt {{{2GM} \over {{R^2}}}}

B) \sqrt {{{2GM} \over R}}

C) \sqrt {{{2gM} \over {{R^2}}}}

D) \sqrt {2g{R^2}}

425

MediumNEET2011

A planet moving along an elliptical orbit is closest to the sun at a distance r1 and farthest away at a distance of r2. If $v1 and v2 are the linear velocities at these points respectively, then the ratio {{{v_1}} \over {{v_2}}}$ is

Options:

A) (r1 /r2)2

B) r2/r1

C) (r2/r1)2

D) r1/r2

426

MediumNEET2010

The dependence of acceleration due to gravity g on the distance r from the centre of the earth, assumed to be a sphere of radius R of uniform density is as shown in figures below The correct figure is

Options:

A) (4)

B) (1)

C) (2)

D) (3)

427

MediumNEET2010

(1) Centre of gravity (C.G) of a body is the point at which the weight of the body acts (2) Centre of mass coincides with the centre of gravity if the earth is assumed to have infinitely large radius. (3) To evaluate the gravitational field intensity due to any body at an external point, the entire mass of the body can be considered to be concentrated at its C.G. (4) The radius of gyration of any body rotating about an axis is the length of the perpendicular dropped from the C.G. of the body to the axis. Which one of the following pairs of statements is correct ?

Options:

A) (4) and (1)

B) (1) and (2)

C) (2) and (3)

D) (3) and (4)

428

MediumNEET2010

The additional kinetic energy to be provided to a satellite of mass m revolving around a planet of mass M, to transfer it from a circular orbit of radius R1 to another of radius R2(R2 > R1) is

Options:

A) GmM $\left( {{1 \over {R_1^2}} - {1 \over {R_2^2}}} \right)

B) GmM $\left( {{1 \over {{R_1}}} - {1 \over {{R_2}}}} \right)

C) 2GmM $\left( {{1 \over {{R_1}}} - {1 \over {{R_2}}}} \right)

D) {1 \over 2}GmM \left( {{1 \over {{R_1}}} - {1 \over {{R_2}}}} \right)

429

MediumNEET2010

A man of 50 kg mass is standing in a gravity free space at a height of 10 m above the floor. He throws a stone of 0.5 kg mass downwards with a speed 2 m/s. When the stone reaches the floor, the distance of the man above the floor will be

Options:

A) 9.9 m

B) 10.1 m

C) 10 m

D) 20 m

430

MediumNEET2010

A particle of mass M is situated at the centre of a spherical shell of same mass and radius a. The magnitude of the gravitational potential at a point sutuated at a/2 distance from the centre, will be :

Options:

A) {{GM} \over a}

B) {{2GM} \over a}

C) {{3GM} \over a}

D) {{4GM} \over a}

431

MediumNEET2010

The radii of circular orbits of two satellites A and B of the earth, are 4R and R, respectively. If the speed of satellite A is 3V, then the speed of satellite B will be

Options:

A) {{3V} \over 4}

B) 6$V

C) 12$V

D) {{3V} \over 2}

432

MediumNEET2009

The figure shows elliptical orbit of a planet m about the sun S. The shaded area SCD is twice the shaded area SAB. If t1 is the time for the planet to move from C to D and t2 is the time to move from A to B then

Options:

A) t1 = 4t2

B) t1 = 2t2

C) t1 = t2

D) t1 > t2

433

MediumNEET2007

Two satellites of earth, S1 and S2 are moving in the same orbit. The mass of S1 is four times the mass of S2. Which one of the following statements is true ?

Options:

A) The potential energies of earth and satellite in the two cases are equal.

B) S1 and S2 are moving with the same speed.

C) The kinetic energies of the two satellites are equal.

D) The time period of S1 is four times that of S2.

434

MediumNEET2006

The earth is assumed to be a sphere of radius R. A platform is arranged at a height R from the surface of the earth. The escape velocity of a body from this platform is fv, where v is its escape velocity from the surface of the Earth. The value of f is

Options:

A) 1/2

B) \sqrt 2

C) 1/$\sqrt 2

D) 1/3

435

MediumNEET2005

For a satellite moving in an orbit around the earth, the ratio of kinetic energy to potential energy is

Options:

A) 1/2

B) 1/$\sqrt 2

C) 2

D) \sqrt 2

436

MediumNEET2005

Imagine a new planet having the same density as that of earth but it is 3 times bigger than the earth in size. If the acceleration due to gravity on the surface of earth is g and that on the surface of the new planet is g', then

Options:

A) g' = g/9

B) g' = 27g

C) g' = 9g

D) g' = 3g

437

MediumNEET2004

The density of a newly discovered planet is twice that of earth. The acceleration due to gravity at the surface of the planet is equal to that at the surface of the earth. If the radius of the earth is R, the radius of the planet would be

Options:

A) 2R

B) 4R

C) {1 \over 4}$R

D) {1 \over 2}$R

438

MediumNEET2003

Two spheres of masses m and M are situated in air and the gravitational force between them is F. The space around the masses is now filled with a liquid of specific gravity 3. The gravitational force will now be

Options:

A) 3 F

B) F

C) F/3

D) F/9

439

MediumNEET2003

The acceleration due to gravity on the planet A is 9 times the acceleration due to gravity on planet B. A man jumps to a height of 2 m on the surface of A. What is the height of jump by the same person on the planet B?

Options:

A) (2/9) m

B) 18 m

C) 6 m

D) (2/3) m

440

MediumNEET2003

A body of mass m is placed on earth's surface which is taken from earth surface to a height of h = 3R, then change in gravitational potential energy is

Options:

A) {{mgR} \over 4}

B) {2 \over 3}mgR

C) {3 \over 4}mgR

D) {{mgR} \over 2}

441

MediumNEET2001

With what velocity should a particle be projected so that its height becomes equal to radius of earth?

Options:

A) {\left( {{{GM} \over R}} \right)^{1/2}}

B) {\left( {{{8GM} \over R}} \right)^{1/2}}

C) {\left( {{{2GM} \over R}} \right)^{1/2}}

D) {\left( {{{4GM} \over R}} \right)^{1/2}}

442

MediumNEET2000

A body of weight 72 N moves from the surface of earth at a height half of the radius of earth, then gravitational force exerted on it will be

Options:

A) 36 N

B) 32 N

C) 144 N

D) 50 N

443

MediumNEET2000

For a planet having mass equal to mass of the earth but radius is one fourth of radius of the earth. Then escape velocity for this planet will be

Options:

A) 11.2 km/sec

B) 22.4 km/sec

C) 5.6 km/sec

D) 44.8 km/sec.

444

MediumNEET2000

Gravitational force is required for

Options:

A) stirring of liquid

B) convection

C) conduction

D) radiation.

445

MediumVITEEE2024

If gravitational attraction between two points masses be given by F=G \frac{m_1 m_2}{r^n}, then the period of a satellite in a circular orbit will be proportional to

Options:

A) r^{\frac{n-1}{2}}

B) r^{\frac{n+1}{2}}

C) r^{\frac{n}{2}}

D) independent of n

446

MediumVITEEE2024

The distance of the centres of Moon and the Earth is D. The mass of the Earth is 81 times the mass of the Moon. At what distance from the centre of the Earth, the gravitational force on a particle will be zero?

Options:

A) \frac{D}{2}

B) \frac{2 D}{3}

C) \frac{4 D}{3}

D) \frac{9 D}{10}

447

MediumVITEEE2023

The gravitational field in a region is given by $\mathbf{E}=5 \mathrm{~N} / \mathrm{kg} \hat{\mathbf{i}}+12 \mathrm{~N} / \mathrm{kg} \hat{\mathbf{j}}. The change in the gravitational potential energy of a particle of mass 1 \mathrm{~kg} when it is taken from the origin to a point (5 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}$) is

Options:

A) 71 \mathrm{~J}

B) 13 \sqrt{58} \mathrm{~J}

C) -71 \mathrm{~J}

D) 35 \mathrm{~J}

448

MediumVITEEE2023

A geostationary satellite is orbiting the Earth at a height of $4 R above that surface of the Earth. R being the radius of the earth. The, time period of another stellite in coins at a height of 2 R$ from the surface of the Earth is.

Options:

A) 2 T_1

B) 2 \sqrt{2} T_1

C) T_1 / 2

D) T_1 / 2 \sqrt{2}

449

MediumVITEEE2022

A skylab or mass $m \mathrm{~kg} is first launched from the surface of the earth in a circular orbit of radius 2 R (from the centre of the earth) and then it is shifted from this circular orbit to another circular orbit of radius 3 R$. The minimum energy required to place the lab in the first orbit and to shift the lab from first orbit to the second orbit are

Options:

A) \frac{3}{4} m g R, \frac{m g R}{6}

B) \frac{3}{4} m g R, \frac{m g R}{12}

C) m g R, m g R

D) 2 m g R, m g R

449

Total Questions

Easy

365

Medium

Hard

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