Class 11 • Physics

Work, Energy and Power & Circular Motion

Chapter-5

310 Questions

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Medium

An electron is moving in a circle of radius 2m with speed of 4 m/s. Find the acceleration of the electron.

Options:

A) 8 m/s$^2

B) 4 m/s$^2

C) 16 m/s$^2

D) 10 m/s$^2

Hard

A bead P sliding on a frictionless semi-circular string (A C B) and it is at point S at t =0 and at this instant the horizontal component of its velocity is v. Another bead Q of the same mass as P is ejected from point A at t=0 along the horizontal string A B, with the speed v, friction between the beads and the respective strings may be neglected in both cases. Let t_P and t_Q be the respective times taken by beads P and Q to reach the point B, then the relation between t_P and t_Q is

Options:

A) t_P < t_Q

B) t_P=t_Q

C) t_P>1.25 t_Q

D) t_P>t_Q

Medium

The particle of mass $m is moving in a circular path of constant radius r such that its centripetal acceleration a_c is varying with time t as a_c=k^2 r t^2, where k$ is a constant. The power delivered to particle by the forces acting on it is

Options:

A) 2 \pi m k^2 r^2 t

B) m k^2 r^2 t

C) 1 / 3 m k^4 r^2 t^5

D) zero

Medium

An object is projected with kinetic energy K from a point A at an angle 60^{\circ} with the horizontal. The ratio of the difference in kinetic energies at points B and C to that at point A (see figure), in the absence of air friction is :

Options:

A) 3: 4

B) 1: 4

C) 2: 3

D) 1: 2

Medium

Assertion : For looping a vertical loop of radius, $r the minimum velocity at lowest point should be \sqrt{5 g r}$. Reason : In this event the velocity at the highest point will be zero.

Options:

A) Both assertion and reason are true and reason is the correct explanation of assertion

B) Both assertion and reason are true but reason is not the correct explanation of assertion

C) Assertion is true but reason is false

D) Both assertion and reason are false.

Medium

Two blocks with masses 100 g and 200 g are attached to the ends of springs A and B as shown in figure. The energy stored in A is E. The energy stored in B, when spring constants k_A, k_B of A and B, respectively satisfy the relation 4 k_A=3 k_B, is :

Options:

A) 4 E

B) 2 E

C) \frac{4}{3} E

D) 3 E

Medium

A cyclist speeding at 6 m/s in a circle of 18 m radius makes an angle $\theta$ with the vertical. The minimum possible value of coefficient of friction between the tyres and the ground is

Options:

A) 12.041

B) 0.2041

C) 11.32

D) 10.020

Easy

Given below are two statements : Statement I : An object moves from position r_1 to position r_2 under a conservative force field \vec{F}. The work done by the force is W=-\int\limits_{r_1}^{r_2} \vec{F} \cdot \overrightarrow{d r}. Statement II : Any object moving from one location to another location can follow infinite number of paths. Therefore, the amount of work done by the object changes with the path it follows for a conservative force. 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 false

D) Both Statement I and Statement II are true

Medium

A point marked on a ring of radius 2 cm is in contact with a horizontal plane. Now the ring is rolled forward half a revolution along the positive X - direction. Then the angle made by the displacement vector of the point with the X - axis is:

Options:

A) \theta=\boldsymbol{\operatorname { t a n }}^{-\mathbf{1}}\left(\frac{\mathbf{2}}{\mathbf{3} \boldsymbol{\pi}}\right)

B) \boldsymbol{\theta}=\boldsymbol{\operatorname { t a n }}^{-\mathbf{1}}\left(\frac{\mathbf{2}}{\boldsymbol{\pi}}\right)

C) \theta=\tan ^{-1}\left(\frac{2 \pi}{3}\right)

D) \theta=\cot ^{-1}\left(\frac{2}{\pi}\right)

Easy

A body of mass 2 kg is moving along x-direction such that its displacement as function of time is given by x(t) = \alpha t^2 + \beta t + \gamma m, where \alpha = 1 \; m/s^2, \beta = 1 \; m/s and \gamma = 1 \; m. The work done on the body during the time interval t = 2 \; s to t = 3 \; s, is ________ J.

Options:

A) 42

B) 24

C) 49

D) 12

Medium

A coin is placed on a disc rotating with an angular velocity \omega. The co-efficient of friction between the disc and the coin is \mu. The maximum distance of the coin from the centre of the disc up to which it will rotate with the disc is

Options:

A) \sqrt{\frac{\mu}{\omega^2}}

B) \frac{\mu g}{\omega^2}

C) \sqrt{\frac{\mu g}{\omega^2}}

D) \frac{\mu g}{\omega}

Medium

Potential energy ( V ) versus distance ( x ) is given by the graph. Rank various regions as per the magnitudes of the force ( F ) acting on a particle from high to low.

Options:

A) F_{B C}>F_{A B}>F_{D E}>F_{C D}

B) F_{B C}>F_{C D}>F_{D E}>F_{A B}

C) F_{C D}>F_{A B}>F_{B C}>F_{D E}

D) F_{C D}>F_{D E}>F_{A B}>F_{B C}

Medium

A ball is moving in a circular path of radius $5 \mathrm{~m}. If tangential acceleration at any instant is 10 \mathrm{~ms}^{-2} and the net acceleration makes an angle of 30^{\circ}$ with the centripetal acceleration, then, the instantaneous speed is

Options:

A) 5.4 \mathrm{~ms}^{-1}

B) 50 \sqrt{3} \mathrm{~ms}^{-1}

C) 6.6 \mathrm{~ms}^{-1}

D) 9.3 \mathrm{~ms}^{-1}

Easy

Which one of the following forces cannot be expressed in terms of potential energy?

Options:

A) Frictional force

B) Coulomb’s force

C) Restoring force

D) Gravitational force

Medium

A metal ball of $20 \mathrm{~g} is projected at an angle 30^{\circ} with the horizontal with an initial velocity 10 \mathrm{~ms}^{-1}$. If the mass and angle of projection are doubled keeping the initial velocity the same, the ratio of the maximum height attained in the former to the latter case is :

Options:

A) 1 : 2

B) 2 : 1

C) 1 : 3

D) 3 : 1

Medium

An object of mass 1000 g experiences a time dependent force \vec{F}=\left(2 t \hat{i}+3 t^2 \hat{j}\right) N. The power generated by the force at time t is:

Options:

A) \left(2 t^2+18 t^3\right) W

B) \left(3 t^3+5 t^5\right) w

C) \left(2 t^3+3 t^5\right) w

D) \left(2 t^2+3 t^3\right) w

Medium

A body is moving along a circular path of radius '$r' with a frequency of revolution numerically equal to the radius of the circular path. What is the acceleration of the body if radius of the path is \left(\frac{5}{\pi}\right) m$ ?

Options:

A) 100 \pi \mathrm{~ms}^{-2}

B) 500 \pi \mathrm{~ms}^{-2}

C) 25 \pi \mathrm{~ms}^{-2}

D) \left(\frac{500}{\pi}\right) \mathrm{ms}^{-2}

Medium

A block of mass 25 kg is pulled along a horizontal surface by a force at an angle 45^{\circ} with the horizontal. The friction coefficient between the block and the surface is 0.25 . The block travels at a uniform velocity. The work done by the applied force during a displacement of 5 m of the block is :

Options:

A) 490 J

B) 970 J

C) 735 J

D) 245 J

Medium

One end of the string of length l is connected to a particle of mass $m and the other end is connected to a small peg on a smooth horizontal table. If the particle moves in circle with speed v, the net force on the particle (directed towards centre) will be ( T$ represents the tension in the string)

Options:

A) T

B) T+\frac{m v^2}{l}

C) T-\frac{m v^2}{l}

D) zero

Easy

A particle is released from height S above the surface of the earth. At certain height its kinetic energy is three times its potential energy. The height from the surface of the earth and the speed of the particle at that instant are respectively.

Options:

A) \frac{\mathrm{S}}{4}, \frac{3 \mathrm{gS}}{2}

B) \frac{\mathrm{S}}{2}, \frac{3 \mathrm{gS}}{2}

C) \frac{\mathrm{S}}{4}, \sqrt{\frac{3 \mathrm{gS}}{2}}

D) \frac{\mathrm{S}}{2}, \sqrt{\frac{3 \mathrm{gS}}{2}}

Medium

When a car of mass $m is moving with speed v along a circle of radius r on a level road, the centripetal force is provided by f, where f denotes (\mu_s \to coefficient of friction, N \to$ normal reaction)

Options:

A) {{m{v^2}} \over r} = f \le {\mu _s}N

B) f < {\mu _s} = {{m{v^2}} \over r}

C) f = {\mu _s}N = {{m{v^2}} \over r}

D) f = {\mu _k}N = {{m{v^2}} \over r}

Medium

A sand dropper drops sand of mass m(t) on a conveyer belt at a rate proportional to the square root of speed (v) of the belt, i.e., \frac{dm}{dt} \propto \sqrt{v}. If P is the power delivered to run the belt at constant speed then which of the following relationship is true?

Options:

A) P \propto \sqrt{v}

B) P \propto v

C) P^2 \propto v^3

D) P^2 \propto v^5

Medium

A cyclist speeding at 6 m/s in a circle of 18 m radius makes an angle $\theta$ with the vertical. The minimum possible value of coefficient of friction between the tyres and the ground is

Options:

A) 12.041

B) 0.2041

C) 11.32

D) 10.020

Medium

A body of mass 4 kg is placed on a plane at a point P having coordinate (3,4) \mathrm{m}. Under the action of force \overrightarrow{\mathrm{F}}=(2 \hat{i}+3 \hat{j}) \mathrm{N}, it moves to a new point Q having coordinates (6,10) \mathrm{m} in 4 sec . The average power and instanteous power at the end of 4 sec are in the ratio of :

Options:

A) 4 : 3

B) 13 : 6

C) 1 : 2

D) 6 : 13

Medium

In case of vertical circular motion of a particle by a thread of length r if the tension in the thread is zero at an angle 30^{\circ} shown in figure, the velocity at the bottom point (A) of the circular path is (g = gravitational acceleration)

Options:

A) \sqrt{\frac{7}{2} g r}

B) \sqrt{4 g r}

C) \sqrt{\frac{5}{2} g r}

D) \sqrt{5 g r}

Medium

A bead of mass ' m ' slides without friction on the wall of a vertical circular hoop of radius ' R ' as shown in figure. The bead moves under the combined action of gravity and a massless spring (k) attached to the bottom of the hoop. The equilibrium length of the spring is ' R '. If the bead is released from top of the hoop with (negligible) zero initial speed, velocity of bead, when the length of spring becomes ' R ', would be (spring constant is ' k ', g is accleration due to gravity)

Options:

A) \sqrt{2 R g+\frac{\mathrm{kR}^2}{\mathrm{~m}}}

B) \sqrt{3 \mathrm{Rg}+\frac{\mathrm{kR}^2}{\mathrm{~m}}}

C) \sqrt{2 \mathrm{Rg}+\frac{4 \mathrm{kR}^2}{\mathrm{~m}}}

D) 2\sqrt{\mathrm{gR}+\frac{\mathrm{kR}^2}{\mathrm{~m}}}

Medium

A large drum having radius R is spinning around its axis with angular velocity \omega, as shown in figure. The minimum value of \omega so that a body of mass M remains stuck to the inner wall of the drum, taking the coefficient of friction between the drum surface and mass M as \mu, is :

Options:

A) \sqrt{\dfrac{2g}{\mu R}}

B) \sqrt{\dfrac{g}{2\mu R}}

C) \sqrt{\dfrac{\mu g}{R}}

D) \sqrt{\dfrac{g}{\mu R}}

Medium

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason \mathbf{R} Assertion A: In a central force field, the work done is independent of the path chosen. Reason R: Every force encountered in mechanics does not have an associated potential energy. In the light of the above statements, choose the most appropriate 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 but \mathbf{R} is NOT 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 and \mathbf{R} is the correct explanation of \mathbf{A}

Easy

Two cars A and B each of mass 10^3 kg are moving on parallel tracks separated by a distance of 10 m, in same direction with speeds 72 km/h and 36 km/h. The magnitude of angular momentum of car A with respect to car B is ________ J·s.

Options:

A) 3.6 \times 10^{5}

B) 10^{5}

C) 3 \times 10^{5}

D) 2 \times 10^{5}

Easy

A force \mathrm{F}=\alpha+\beta \mathrm{x}^2 acts on an object in the x -direction. The work done by the force is 5 J when the object is displaced by 1 m . If the constant \alpha=1 \mathrm{~N} then \beta will be

Options:

A) 15 \mathrm{~N} / \mathrm{m}^2

B) 10 \mathrm{~N} / \mathrm{m}^2

C) 12 \mathrm{~N} / \mathrm{m}^2

D) 8 \mathrm{~N} / \mathrm{m}^2

Medium

A wheel is rolling on a plane surface. The speed of a particle on the highest point of the rim is 8 \mathrm{~m} / \mathrm{s}. The speed of the particle on the rim of the wheel at the same level as the centre of wheel, will be :

Options:

A) 4 \sqrt{2} \mathrm{~m} / \mathrm{s}

B) 8 \mathrm{~m} / \mathrm{s}

C) 4 \mathrm{~m} / \mathrm{s}

D) 8 \sqrt{2} \mathrm{~m} / \mathrm{s}

Easy

A ball having kinetic energy KE, is projected at an angle of 60^{\circ} from the horizontal. What will be the kinetic energy of ball at the highest point of its flight?

Options:

A) \frac{(\mathrm{KE})}{2}

B) \frac{(\mathrm{KE})}{8}

C) \frac{(\mathrm{KE})}{4}

D) \frac{(\mathrm{KE})}{16}

Easy

A sportsman runs around a circular track of radius r such that he traverses the path A B A B. The distance travelled and displacement, respectively, are

Options:

A) \pi r, 3 r

B) 2 \mathrm{r}, 3 \pi \mathrm{r}

C) 3 \pi \mathrm{r}, 2 \mathrm{r}

D) 3 \pi r, \pi r

Easy

A force \overrightarrow{\mathrm{F}}=2 \hat{i}+\mathrm{b} \hat{j}+\hat{k} is applied on a particle and it undergoes a displacement \hat{i}-2 \hat{j}-\hat{k} What will be the value of b, if work done on the particle is zero.

Options:

A) \frac{1}{3}

B) \frac{1}{2}

C) 0

D) 2

Medium

A body of mass ‘m’ connected to a massless and unstretchable string goes in vertical circle of radius ‘R’ under gravity g. The other end of the string is fixed at the center of circle. If velocity at top of circular path is n\sqrt{ g R} , where, n ≥ 1, then ratio of kinetic energy of the body at bottom to that at top of the circle is :

Options:

A) \frac{n^2 + 4}{n^2}

B) \frac{n + 4}{n}

C) \frac{n^2}{n^2 + 4}

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

Easy

A particle of mass $m moves on a straight line with its velocity increasing with distance according to the equation v=\alpha \sqrt{x}, where \alpha is a constant. The total work done by all the forces applied on the particle during its displacement from x=0 to x=\mathrm{d}$, will be :

Options:

A) \frac{\mathrm{m}}{2 \alpha^2 \mathrm{~d}}

B) \frac{\mathrm{md}}{2 \alpha^2}

C) \frac{\mathrm{m} \alpha^2 \mathrm{~d}}{2}

D) 2 \mathrm{~m} \alpha^2 \mathrm{~d}

Medium

A car of mass ' m ' moves on a banked road having radius ' r ' and banking angle \theta. To avoid slipping from banked road, the maximum permissible speed of the car is v_0. The coefficient of friction \mu between the wheels of the car and the banked road is

Options:

A) \mu=\frac{v_0^2+r g \tan \theta}{r g+v_0^2 \tan \theta}

B) \mu=\frac{v_0^2-r g \tan \theta}{\mathrm{rg}-\mathrm{v}_{\mathrm{o}}^2 \tan \theta}

C) \mu=\frac{v_0^2-r g \tan \theta}{r g+v_0^2 \tan \theta}

D) \mu=\frac{v_o^2+r g \tan \theta}{r g-v_o^2 \tan \theta}

Medium

A block is simply released from the top of an inclined plane as shown in the figure above. The maximum compression in the spring when the block hits the spring is :

Options:

A) \sqrt{6} \mathrm{~m}

B) \sqrt{5} \mathrm{~m}

C) 1 \mathrm{~m}

D) 2 \mathrm{~m}

Medium

A body of mass 100 g is moving in circular path of radius 2 m on vertical plane as shown in figure. The velocity of the body at point A is 10 \mathrm{~m} / \mathrm{s}. The ratio of its kinetic energies at point B and C is : (Take acceleration due to gravity as 10 \mathrm{~m} / \mathrm{s}^2)

Options:

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

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

C) \frac{2+\sqrt{2}}{3}

D) \frac{3+\sqrt{3}}{2}

Easy

Three bodies A, B and C have equal kinetic energies and their masses are $400 \mathrm{~g}, 1.2 \mathrm{~kg} and 1.6 \mathrm{~kg}$ respectively. The ratio of their linear momenta is :

Options:

A) 1: \sqrt{3}: 2

B) \sqrt{3}: \sqrt{2}: 1

C) 1: \sqrt{3}: \sqrt{2}

D) \sqrt{2} : \sqrt{3}: 1

Medium

A bob of mass m is suspended at a point O by a light string of length l and left to perform vertical motion (circular) as shown in figure. Initially, by applying horizontal velocity v_0 at the point ' A ', the string becomes slack when, the bob reaches at the point ' D '. The ratio of the kinetic energy of the bob at the points B and C is _________.

Options:

A) 4

B) 1

C) 2

D) 3

Easy

When kinetic energy of a body becomes 36 times of its original value, the percentage increase in the momentum of the body will be :

Options:

A) 60%

B) 500%

C) 6%

D) 600%

Medium

A clock has $75 \mathrm{~cm}, 60 \mathrm{~cm} long second hand and minute hand respectively. In 30 minutes duration the tip of second hand will travel x distance more than the tip of minute hand. The value of x in meter is nearly (Take \pi=3.14$) :

Options:

A) 118.9

B) 140.5

C) 139.4

D) 220.0

Easy

A bullet of mass $50 \mathrm{~g} is fired with a speed 100 \mathrm{~m} / \mathrm{s} on a plywood and emerges with 40 \mathrm{~m} / \mathrm{s}$. The percentage loss of kinetic energy is :

Options:

A) 44 \%

B) 16 \%

C) 84 \%

D) 32 \%

Medium

A car of $800 \mathrm{~kg} is taking turn on a banked road of radius 300 \mathrm{~m} and angle of banking 30^{\circ}. If coefficient of static friction is 0.2 then the maximum speed with which car can negotiate the turn safely: (\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2, \sqrt{3}=1.73)

Options:

A) 51.4 m/s

B) 102.8 m/s

C) 70.4 m/s

D) 264 m/s

Easy

Four particles $A, B, C, D of mass \frac{m}{2}, m, 2 m, 4 m$, have same momentum, respectively. The particle with maximum kinetic energy is :

Options:

A) B

B) C

C) D

D) A

Easy

A man carrying a monkey on his shoulder does cycling smoothly on a circular track of radius $9 \mathrm{~m} and completes 120 resolutions in 3 minutes. The magnitude of centripetal acceleration of monkey is (in \mathrm{m} / \mathrm{s}^2$ ) :

Options:

A) 4 \pi^2 \mathrm{~ms}^{-2}

B) 16 \pi^2 \mathrm{~ms}^{-2}

C) 57600 \pi^2 \mathrm{~ms}^{-2}

D) Zero

Medium

A body is moving unidirectionally under the influence of a constant power source. Its displacement in time t is proportional to :

Options:

A) t2/3

B) t3/2

C) t

D) t2

Easy

A cyclist starts from the point $P of a circular ground of radius 2 \mathrm{~km} and travels along its circumference to the point \mathrm{S}$. The displacement of a cyclist is:

Options:

A) \sqrt8$ km

B) 4 km

C) 6 km

D) 8 km

Easy

A body of mass $50 \mathrm{~kg} is lifted to a height of 20 \mathrm{~m}$ from the ground in the two different ways as shown in the figures. The ratio of work done against the gravity in both the respective cases, will be :

Options:

A) 2: 1

B) \sqrt{3}: 2

C) 1: 1

D) 1: 2

Easy

A ball of mass 0.5 \mathrm{~kg} is attached to a string of length 50 \mathrm{~cm}. The ball is rotated on a horizontal circular path about its vertical axis. The maximum tension that the string can bear is 400 \mathrm{~N}. The maximum possible value of angular velocity of the ball in \mathrm{rad} / \mathrm{s} is, :

Options:

A) 1600

B) 20

C) 40

D) 1000

Medium

A body of $m \mathrm{~kg} slides from rest along the curve of vertical circle from point A to B in friction less path. The velocity of the body at B is: (given, R=14 \mathrm{~m}, g=10 \mathrm{~m} / \mathrm{s}^2 and \sqrt{2}=1.4$)

Options:

A) 10.6 m/s

B) 19.8 m/s

C) 16.7 m/s

D) 21.9 m/s

Medium

A particle moving in a circle of radius \mathrm{R} with uniform speed takes time \mathrm{T} to complete one revolution. If this particle is projected with the same speed at an angle \theta to the horizontal, the maximum height attained by it is equal to 4 R. The angle of projection \theta is then given by :

Options:

A) \sin ^{-1}\left[\frac{2 \mathrm{gT}^2}{\pi^2 \mathrm{R}}\right]^{\frac{1}{2}}

B) \sin ^{-1}\left[\frac{\pi^2 \mathrm{R}}{2 \mathrm{gT}^2}\right]^{\frac{1}{2}}

C) \cos ^{-1}\left[\frac{\pi \mathrm{R}}{2 \mathrm{gT}^2}\right]^{\frac{1}{2}}

D) \cos ^{-1}\left[\frac{2 \mathrm{gT}^2}{\pi^2 \mathrm{R}}\right]^{\frac{1}{2}}

Easy

If a rubber ball falls from a height $h and rebounds upto the height of h / 2$. The percentage loss of total energy of the initial system as well as velocity ball before it strikes the ground, respectively, are :

Options:

A) 50 \%, \sqrt{2 \mathrm{gh}}

B) 50 \%, \sqrt{\mathrm{gh}}

C) 50 \%, \sqrt{\frac{\text { gh }}{2}}

D) 40 \%, \sqrt{2 \mathrm{gh}}

Easy

A coin is placed on a disc. The coefficient of friction between the coin and the disc is $\mu. If the distance of the coin from the center of the disc is r$, the maximum angular velocity which can be given to the disc, so that the coin does not slip away, is :

Options:

A) \sqrt{\frac{r}{\mu g}}

B) \sqrt{\frac{\mu g}{r}}

C) \frac{\mu g}{r}

D) \frac{\mu}{\sqrt{r g}}

Medium

A body of mass $2 \mathrm{~kg} begins to move under the action of a time dependent force given by \vec{F}=\left(6 t \hat{i}+6 t^2 \hat{j}\right) N. The power developed by the force at the time t$ is given by:

Options:

A) \left(3 t^3+6 t^5\right) W

B) \left(9 t^5+6 t^3\right) W

C) \left(6 t^4+9 t^5\right) W

D) \left(9 t^3+6 t^5\right) W

Medium

A stone of mass $900 \mathrm{~g} is tied to a string and moved in a vertical circle of radius 1 \mathrm{~m} making 10 \mathrm{~rpm}. The tension in the string, when the stone is at the lowest point is (if \pi^2=9.8 and g=9.8 \mathrm{~m} / \mathrm{s}^2$) :

Options:

A) 17.8 N

B) 97 N

C) 9.8 N

D) 8.82 N

Medium

A block of mass $1 \mathrm{~kg} is pushed up a surface inclined to horizontal at an angle of 60^{\circ} by a force of 10 \mathrm{~N} parallel to the inclined surface as shown in figure. When the block is pushed up by 10 \mathrm{~m} along inclined surface, the work done against frictional force is : \left[g=10 \mathrm{~m} / \mathrm{s}^2\right]

Options:

A) 5$\sqrt3$ J

B) 5 J

C) 5\times10^3$ J

D) 10 J

Easy

If the radius of curvature of the path of two particles of same mass are in the ratio $3: 4$, then in order to have constant centripetal force, their velocities will be in the ratio of :

Options:

A) 1: \sqrt{3}

B) 2: \sqrt{3}

C) \sqrt{3}: 2

D) \sqrt{3}: 1

Easy

A particle is placed at the point $A of a frictionless track A B C as shown in figure. It is gently pushed towards right. The speed of the particle when it reaches the point B is : (Take g=10 \mathrm{~m} / \mathrm{s}^2$).

Options:

A) 2 \sqrt{10} \mathrm{~m} / \mathrm{s}

B) 10 \mathrm{~m} / \mathrm{s}

C) \sqrt{10} \mathrm{~m} / \mathrm{s}

D) 20 \mathrm{~m} / \mathrm{s}

Easy

A train is moving with a speed of $12 \mathrm{~m} / \mathrm{s} on rails which are 1.5 \mathrm{~m} apart. To negotiate a curve radius 400 \mathrm{~m}, the height by which the outer rail should be raised with respect to the inner rail is (Given, g=10 \mathrm{~m} / \mathrm{s}^2)$ :

Options:

A) 6.0 cm

B) 5.4 cm

C) 4.8 cm

D) 4.2 cm

Medium

A bob of mass '$m' is suspended by a light string of length 'L'. It is imparted a minimum horizontal velocity at the lowest point A such that it just completes half circle reaching the top most position B. The ratio of kinetic energies \frac{(K . E)_A}{(K . E)_B}$ is :

Options:

A) 5 : 1

B) 3 : 2

C) 1 : 5

D) 2 : 5

Easy

A vehicle of mass $200 \mathrm{~kg} is moving along a levelled curved road of radius 70 \mathrm{~m} with angular velocity of 0.2 ~\mathrm{rad} / \mathrm{s}$. The centripetal force acting on the vehicle is:

Options:

A) 560 \mathrm{~N}

B) 14 \mathrm{~N}

C) 2800 \mathrm{~N}

D) 2240 \mathrm{~N}

Easy

A block of mass $100 \mathrm{~kg} slides over a distance of 10 \mathrm{~m} on a horizontal surface. If the co-efficient of friction between the surfaces is 0.4, then the work done against friction (\operatorname{in} J$) is :

Options:

A) 3900

B) 4500

C) 4200

D) 4000

Easy

A coin placed on a rotating table just slips when it is placed at a distance of $1 \mathrm{~cm}$ from the center. If the angular velocity of the table in halved, it will just slip when placed at a distance of _________ from the centre :

Options:

A) 1 cm

B) 8 cm

C) 4 cm

D) 2 cm

Easy

The potential energy function (in $J ) of a particle in a region of space is given as U=\left(2 x^2+3 y^3+2 z\right). Here x, y and z are in meter. The magnitude of x-component of force (in N ) acting on the particle at point P(1,2,3) \mathrm{m}$ is :

Options:

A) 4

B) 2

C) 8

D) 6

Medium

As shown in the figure, a particle is moving with constant speed $\pi ~\mathrm{m} / \mathrm{s}. Considering its motion from \mathrm{A} to \mathrm{B}$, the magnitude of the average velocity is :

Options:

A) \pi ~\mathrm{m} / \mathrm{s}

B) 1.5 \sqrt{3} \mathrm{~m} / \mathrm{s}

C) \sqrt{3} \mathrm{~m} / \mathrm{s}

D) 2 \sqrt{3} \mathrm{~m} / \mathrm{s}

Medium

A ball suspended by a thread swings in a vertical plane so that its magnitude of acceleration in the extreme position and lowest position are equal. The angle $(\theta)$ of thread deflection in the extreme position will be :

Options:

A) \tan ^{-1}\left(\frac{1}{2}\right)

B) 2 \tan ^{-1}\left(\frac{1}{2}\right)

C) 2 \tan ^{-1}\left(\frac{1}{\sqrt{5}}\right)

D) \tan ^{-1}(\sqrt{2})

Easy

A child of mass $5 \mathrm{~kg} is going round a merry-go-round that makes 1 rotation in 3.14 \mathrm{~s}. The radius of the merry-go-round is 2 \mathrm{~m}$. The centrifugal force on the child will be

Options:

A) 50 N

B) 80 N

C) 100 N

D) 40 N

Medium

A bullet is fired into a fixed target looses one third of its velocity after travelling $4 \mathrm{~cm}. It penetrates further \mathrm{D} \times 10^{-3} \mathrm{~m} before coming to rest. The value of \mathrm{D}$ is :

Options:

A) 23

B) 32

C) 42

D) 52

Medium

A particle is moving with constant speed in a circular path. When the particle turns by an angle $90^{\circ}, the ratio of instantaneous velocity to its average velocity is \pi: x \sqrt{2}. The value of x$ will be -

Options:

A) 1

B) 7

C) 5

D) 2

Medium

The ratio of powers of two motors is $\frac{3 \sqrt{x}}{\sqrt{x}+1}, that are capable of raising 300 \mathrm{~kg} water in 5 minutes and 50 \mathrm{~kg} water in 2 minutes respectively from a well of 100 \mathrm{~m} deep. The value of x$ will be

Options:

A) 16

B) 4

C) 2

D) 2.4

Medium

A small block of mass $100 \mathrm{~g} is tied to a spring of spring constant 7.5 \mathrm{~N} / \mathrm{m} and length 20 \mathrm{~cm}. The other end of spring is fixed at a particular point A. If the block moves in a circular path on a smooth horizontal surface with constant angular velocity 5 ~\mathrm{rad} / \mathrm{s} about point \mathrm{A}$, then tension in the spring is -

Options:

A) 0.50 N

B) 1.5 N

C) 0.75 N

D) 0.25 N

Easy

Two bodies are having kinetic energies in the ratio 16 : 9. If they have same linear momentum, the ratio of their masses respectively is :

Options:

A) 3: 4

B) 4: 3

C) 9: 16

D) 16: 9

Medium

A stone of mass 1 \mathrm{~kg} is tied to end of a massless string of length 1 \mathrm{~m}. If the breaking tension of the string is 400 \mathrm{~N}, then maximum linear velocity, the stone can have without breaking the string, while rotating in horizontal plane, is :

Options:

A) 20 \mathrm{~ms}^{-1}

B) 40 \mathrm{~ms}^{-1}

C) 400 \mathrm{~ms}^{-1}

D) 10 \mathrm{~ms}^{-1}

Easy

Given below are two statements: Statement I : A truck and a car moving with same kinetic energy are brought to rest by applying breaks which provide equal retarding forces. Both come to rest in equal distance. Statement II : A car moving towards east takes a turn and moves towards north, the speed remains unchanged. The acceleration of the car is zero. In the light of given statements, choose the most appropriate answer from the options given below

Options:

A) Statement I is incorrect but Statement II is correct.

B) Statement $\mathrm{I}$ is correct but Statement II is incorrect.

C) Both Statement I and Statement II are correct.

D) Both Statement I and Statement II are incorrect.

Easy

A body is moving with constant speed, in a circle of radius 10 \mathrm{~m}. The body completes one revolution in 4 \mathrm{~s}. At the end of 3rd second, the displacement of body (in \mathrm{m} ) from its starting point is :

Options:

A) 15 \pi

B) 30

C) 10 \sqrt{2}

D) 5 \pi

Medium

A bullet of mass $0.1 \mathrm{~kg} moving horizontally with speed 400 \mathrm{~ms}^{-1} hits a wooden block of mass 3.9 \mathrm{~kg} kept on a horizontal rough surface. The bullet gets embedded into the block and moves 20 \mathrm{~m} before coming to rest. The coefficient of friction between the block and the surface is __________. (Given g=10 \mathrm{~m} / \mathrm{s}^{2}$ )

Options:

A) 0.65

B) 0.25

C) 0.50

D) 0.90

Easy

An object moves at a constant speed along a circular path in a horizontal plane with center at the origin. When the object is at $x=+2~\mathrm{m}, its velocity is \mathrm{ - 4\widehat j} m/s. The object's velocity (v) and acceleration (a) at x=-2~\mathrm{m}$ will be

Options:

A) v=4\mathrm{\widehat i~m/s},a=8\mathrm{\widehat j~m/s^2}

B) v=4\mathrm{\widehat j~m/s},a=8\mathrm{\widehat i~m/s^2}

C) v=-4\mathrm{\widehat i~m/s},a=-8\mathrm{\widehat j~m/s^2}

D) v=-4\mathrm{\widehat j~m/s},a=8\mathrm{\widehat i~m/s^2}

Medium

Identify the correct statements from the following : A. Work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket is negative. B. Work done by gravitational force in lifting a bucket out of a well by a rope tied to the bucket is negative. C. Work done by friction on a body sliding down an inclined plane is positive. D. Work done by an applied force on a body moving on a rough horizontal plane with uniform velocity is zero. E. Work done by the air resistance on an oscillating pendulum is negative. Choose the correct answer from the options given below :

Options:

A) A and C only

B) B and D only

C) B, D and E only

D) B and E only

Easy

A car is moving on a horizontal curved road with radius 50 m. The approximate maximum speed of car will be, if friction between tyres and road is 0.34. [take g = 10 ms$^{-2}$]

Options:

A) 3.4 ms$^{-1}

B) 13 ms$^{-1}

C) 22.4 ms$^{-1}

D) 17 ms$^{-1}

Easy

A stone is projected at angle $30^{\circ}$ to the horizontal. The ratio of kinetic energy of the stone at point of projection to its kinetic energy at the highest point of flight will be -

Options:

A) 1 : 4

B) 1 : 2

C) 4 : 3

D) 4 : 1

Easy

A car is moving with a constant speed of 20 m/s in a circular horizontal track of radius 40 m. A bob is suspended from the roof of the car by a massless string. The angle made by the string with the vertical will be : (Take g = 10 m/s$^2$)

Options:

A) \frac{\pi}{2}

B) \frac{\pi}{6}

C) \frac{\pi}{4}

D) \frac{\pi}{3}

Easy

A ball is projected with kinetic energy E, at an angle of $60^{\circ}$ to the horizontal. The kinetic energy of this ball at the highest point of its flight will become :

Options:

A) Zero

B) \frac{E}{2}

C) \frac{E}{4}

D) E

Medium

A body of mass 200g is tied to a spring of spring constant 12.5 N/m, while the other end of spring is fixed at point O. If the body moves about O in a circular path on a smooth horizontal surface with constant angular speed 5 rad/s. Then the ratio of extension in the spring to its natural length will be :

Options:

A) 1 : 2

B) 2 : 3

C) 2 : 5

D) 1 : 1

Medium

A bullet of mass $200 \mathrm{~g} having initial kinetic energy 90 \mathrm{~J} is shot inside a long swimming pool as shown in the figure. If it's kinetic energy reduces to 40 \mathrm{~J} within 1 \mathrm{~s}$, the minimum length of the pool, the bullet has to travel so that it completely comes to rest is

Options:

A) 45 m

B) 90 m

C) 125 m

D) 25 m

Easy

A smooth circular groove has a smooth vertical wall as shown in figure. A block of mass m moves against the wall with a speed v. Which of the following curve represents the correct relation between the normal reaction on the block by the wall (N) and speed of the block (v) ?

Options:

Medium

Sand is being dropped from a stationary dropper at a rate of $0.5 \,\mathrm{kgs}^{-1} on a conveyor belt moving with a velocity of 5 \mathrm{~ms}^{-1}$. The power needed to keep the belt moving with the same velocity will be :

Options:

A) 1.25 W

B) 2.5 W

C) 6.25 W

D) 12.5 W

Medium

A person moved from A to B on a circular path as shown in figure. If the distance travelled by him is $60 \mathrm{~m}, then the magnitude of displacement would be : (Given \left.\cos 135^{\circ}=-0.7\right)

Options:

A) 42 m

B) 47 m

C) 19 m

D) 40 m

Easy

As per the given figure, two blocks each of mass $250 \mathrm{~g} are connected to a spring of spring constant 2 \,\mathrm{Nm}^{-1}. If both are given velocity v$ in opposite directions, then maximum elongation of the spring is :

Options:

A) \frac{v}{2 \sqrt{2}}

B) \frac{v}{2}

C) \frac{v}{4}

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

Medium

A particle of mass m is moving in a circular path of constant radius r such that its centripetal acceleration (a) is varying with time t as a = k2rt2, where k is a constant. The power delivered to the particle by the force acting on it is given as

Options:

A) zero

B) mk2r2t2

C) mk2r2t

D) mk2rt

Easy

A bag of sand of mass 9.8 kg is suspended by a rope. A bullet of 200 g travelling with speed 10 ms$-$1 gets embedded in it, then loss of kinetic energy will be :

Options:

A) 4.9 J

B) 9.8 J

C) 14.7 J

D) 19.6 J

Medium

A stone tide to a spring of length L is whirled in a vertical circle with the other end of the spring at the centre. At a certain instant of time, the stone is at its lowest position and has a speed u. The magnitude of change in its velocity, as it reaches a position where the string is horizontal, is $\sqrt {x({u^2} - gL)} $. The value of x is -

Options:

A) 3

B) 2

C) 1

D) 5

Easy

A body of mass $0.5 \mathrm{~kg} travels on straight line path with velocity v=\left(3 x^{2}+4\right) \mathrm{m} / \mathrm{s}. The net workdone by the force during its displacement from x=0 to x=2 \mathrm{~m}$ is :

Options:

A) 64 J

B) 60 J

C) 120 J

D) 128 J

Medium

A ball is released from rest from point P of a smooth semi-spherical vessel as shown in figure. The ratio of the centripetal force and normal reaction on the ball at point Q is A while angular position of point Q is $\alpha with respect to point P. Which of the following graphs represent the correct relation between A and \alpha$ when ball goes from Q to R?

Options:

Easy

In the given figure, the block of mass m is dropped from the point 'A'. The expression for kinetic energy of block when it reaches point 'B' is

Options:

A) {1 \over 2}mg\,{y_0}^2

B) {1 \over 2}mg\,{y^2}

C) mg(y - {y_0})

D) mg{y_0}

Medium

A disc with a flat small bottom beaker placed on it at a distance R from its center is revolving about an axis passing through the center and perpendicular to its plane with an angular velocity $\omega. The coefficient of static friction between the bottom of the beaker and the surface of the disc is \mu$. The beaker will revolve with the disc if :

Options:

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

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

C) R \ge {{\mu g} \over {2{\omega ^2}}}

D) R \ge {{\mu g} \over {{\omega ^2}}}

Medium

A particle of mass 500 gm is moving in a straight line with velocity v = b x5/2. The work done by the net force during its displacement from x = 0 to x = 4 m is : (Take b = 0.25 m$-3/2 s-$1).

Options:

A) 2 J

B) 4 J

C) 8 J

D) 16 J

Easy

For a particle in uniform circular motion, the acceleration $\overrightarrow a at any point P(R, \theta) on the circular path of radius R is (when \theta$ is measured from the positive x-axis and v is uniform speed) :

Options:

A) - {{{v^2}} \over R}\sin \theta \widehat i + {{{v^2}} \over R}\cos \theta \widehat j

B) - {{{v^2}} \over R}\cos \theta \widehat i + {{{v^2}} \over R}\sin \theta \widehat j

C) - {{{v^2}} \over R}\cos \theta \widehat i - {{{v^2}} \over R}\sin \theta \widehat j

D) - {{{v^2}} \over R}\widehat i + {{{v^2}} \over R}\widehat j

100

Medium

Arrange the four graphs in descending order of total work done; where W1, W2, W3 and W4 are the work done corresponding to figure a, b, c and d respectively.

Options:

A) W3 > W2 > W1 > W4

B) W3 > W2 > W4 > W1

C) W2 > W3 > W4 > W1

D) W2 > W3 > W1 > W4

101

Medium

A stone of mass m, tied to a string is being whirled in a vertical circle with a uniform speed. The tension in the string is

Options:

A) the same throughout the motion.

B) minimum at the highest position of the circular path.

C) minimum at the lowest position of the circular path.

D) minimum when the rope is in the horizontal position.

102

Medium

A particle experiences a variable force $\overrightarrow F = \left( {4x\widehat i + 3{y^2}\widehat j} \right)$ in a horizontal x-y plane. Assume distance in meters and force is newton. If the particle moves from point (1, 2) to point (2, 3) in the x-y plane, then Kinetic Energy changes by :

Options:

A) 50.0 J

B) 12.5 J

C) 25.0 J

D) 0 J

103

Easy

A fly wheel is accelerated uniformly from rest and rotates through 5 rad in the first second. The angle rotated by the fly wheel in the next second, will be :

Options:

A) 7.5 rad

B) 15 rad

C) 20 rad

D) 30 rad

104

Easy

A body of mass 'm' dropped from a height 'h' reaches the ground with a speed of 0.8$\sqrt {gh} $. The value of workdone by the air-friction is :

Options:

A) -$0.68 mgh

B) mgh

C) 1.64 mgh

D) 0.64 mgh

105

Easy

A boy ties a stone of mass 100 g to the end of a 2 m long string and whirls it around in a horizontal plane. The string can withstand the maximum tension of 80 N. If the maximum speed with which the stone can revolve is ${K \over \pi }$ rev./min. The value of K is : (Assume the string is massless and unstretchable)

Options:

A) 400

B) 300

C) 600

D) 800

106

Medium

A block moving horizontally on a smooth surface with a speed of 40 m/s splits into two parts with masses in the ratio of 1 : 2. If the smaller part moves at 60 m/s in the same direction, then the fractional change in kinetic energy is :-

Options:

A) {{1 \over 3}}

B) {{2 \over 3}}

C) {{1 \over 8}}

D) {{1 \over 4}}

107

Medium

A huge circular arc of length 4.4 ly subtends an angle '4s' at the centre of the circle. How long it would take for a body to complete 4 revolution if its speed is 8 AU per second?Given : 1 ly = 9.46 $\times 1015 m1 AU = 1.5 \times$ 1011 m

Options:

A) 4.1 $\times$ 108 s

B) 4.5 $\times$ 1010 s

C) 3.5 $\times$ 106 s

D) 7.2 $\times$ 108 s

108

Medium

Given below is the plot of a potential energy function U(x) for a system, in which a particle is in one dimensional motion, while a conservative force F(x) acts on it. Suppose that Emech = 8 J, the incorrect statement for this system is : [ where K.E. = kinetic energy ]

Options:

A) at x > x4 K.E. is constant throughout the region.

B) at x < x1, K.E. is smallest and the particle is moving at the slowest speed.

C) at x = x2, K.E. is greatest and the particle is moving at the fastest speed.

D) at x = x3, K.E. = 4 J.

109

Medium

A particle of mass m is suspended from a ceiling through a string of length L. The particle moves in a horizontal circle of radius r such that $r = {L \over {\sqrt 2 }}$. The speed of particle will be :

Options:

A) {\sqrt {rg} }

B) {\sqrt {2rg} }

C) {2\sqrt {rg} }

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

110

Medium

An automobile of mass 'm' accelerates starting from origin and initially at rest, while the engine supplies constant power P. The position is given as a function of time by :

Options:

A) {\left( {{{9P} \over {8m}}} \right)^{{1 \over 2}}}{t^{{3 \over 2}}}

B) {\left( {{{8P} \over {9m}}} \right)^{{1 \over 2}}}{t^{{2 \over 3}}}

C) {\left( {{{9m} \over {8P}}} \right)^{{1 \over 2}}}{t^{{3 \over 2}}}

D) {\left( {{{8P} \over {9m}}} \right)^{{1 \over 2}}}{t^{{3 \over 2}}}

111

Medium

The normal reaction 'N' for a vehicle of 800 kg mass, negotiating a turn on a 30$^\circ banked road at maximum possible speed without skidding is ____________ \times 103 kg m/s2. [Given cos30^\circ = 0.87, \mu$s = 0.2]

Options:

A) 12.4

B) 7.2

C) 6.96

D) 10.2

112

Easy

A porter lifts a heavy suitcase of mass 80 kg and at the destination lowers it down by a distance of 80 cm with a constant velocity. Calculate the work done by the porter in lowering the suitcase.(take g = 9.8 ms$-$2)

Options:

A) +627.2 J

B) -$62720.0 J

C) -$627.2 J

D) 784.0 J

113

Medium

A particle of mass m moves in a circular orbit under the central potential field, $U(r) = - {C \over r}, where C is a positive constant. The correct radius -$ velocity graph of the particle's motion is :

Options:

114

Medium

A body at rest is moved along a horizontal straight line by a machine delivering a constant power. The distance moved by the body in time 't' is proportional to :

Options:

A) {t^{{3 \over 2}}}

B) {t^{{1 \over 2}}}

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

D) {t^{{3 \over 4}}}

115

Medium

A modern grand - prix racing car of mass m is travelling on a flat track in a circular arc of radius R with a speed v. If the coefficient of static friction between the tyres and the track is $\mu$s, then the magnitude of negative lift FL acting downwards on the car is : (Assume forces on the four tyres are identical and g = acceleration due to gravity)

Options:

A) m\left( {g - {{{v^2}} \over {{\mu _s}R}}} \right)

B) - m\left( {g + {{{v^2}} \over {{\mu _s}R}}} \right)

C) m\left( {{{{v^2}} \over {{\mu _s}R}} - g} \right)

D) m\left( {{{{v^2}} \over {{\mu _s}R}} + g} \right)

116

Medium

A constant power delivering machine has towed a box, which was initially at rest, along a horizontal straight line. The distance moved by the box in time 't' is proportional to :-

Options:

A) t2/3

B) t3/2

C) t

D) t1/2

117

Medium

Statement I : A cyclist is moving on an unbanked road with a speed of 7 kmh$-1 and takes a sharp circular turn along a path of radius of 2m without reducing the speed. The static friction coefficient is 0.2. The cyclist will not slip and pass the curve. (g = 9.8 m/s2)Statement II : If the road is banked at an angle of 45^\circ, cyclist can cross the curve of 2m radius with the speed of 18.5 kmh-$1 without slipping.In the light of the above statements, choose the correct answer from the options given below.

Options:

A) Statement I is incorrect and statement II is correct

B) Both statement I and statement II are true

C) Statement I is correct and statement II is incorrect

D) Both statement I and statement II are false

118

Easy

A boy is rolling a 0.5 kg ball on the frictionless floor with the speed of 20 ms-1. The ball gets deflected by an obstacle on the way. After deflection it moves with 5% of its initial kinetic energy. What is the speed of the ball now?

Options:

A) 14.41 ms$-$1

B) 19.0 ms$-$1

C) 4.47 ms$-$1

D) 1.00 ms$-$1

119

Easy

A block of 200 g mass moves with a uniform speed in a horizontal circular groove, with vertical side walls of radius 20 cm. If the block takes 40 s to complete one round, the normal force by the side walls of the groove is :

Options:

A) 9.859 $\times 10-$2 N

B) 0.0314 N

C) 9.859 $\times 10-$4 N

D) 6.28 $\times 10-$3 N

120

Medium

If the potential energy between two molecules is given by U = $ - {A \over {{r^6}}} + {B \over {{r^{12}}}}$, then at equilibrium, separation between molecules, and the potential energy are :

Options:

A) {\left( {{{2B} \over A}} \right)^{1/6}}, - {{{A^2}} \over {4B}}

B) {\left( {{{2B} \over A}} \right)^{1/6}}, - {{{A^2}} \over {2B}}

C) {\left( {{B \over A}} \right)^{1/6}},0

D) {\left( {{B \over {2A}}} \right)^{1/6}}, - {{{A^2}} \over {2B}}

121

Medium

A particle is moving with uniform speed along the circumference of a circle of radius R under the action of a central fictitious force F which is inversely proportional to R3. Its time period of revolution will be given by :

Options:

A) T \propto {R^{{4 \over 3}}}

B) T \propto {R^{{5 \over 2}}}

C) T \propto {R^{{3 \over 2}}}

D) T \propto {R^2}

122

Medium

A person pushes a box on a rough horizontal plateform surface. He applies a force of 200 N over a distance of 15 m. Thereafter, he gets progressively tired and his applied force reduces linearly with distance to 100 N. The total distance through which the box has been moved is 30 m. What is the work done by the person during the total movement of the box?

Options:

A) 5690 J

B) 5250 J

C) 2780 J

D) 3280 J

123

Medium

A clock has a continuously moving second's hand of 0.1 m length. The average acceleration of the tip of the hand (in units of ms–2) is of the order of :

Options:

A) 10-3

B) 10-1

C) 10-2

D) 10-4

124

Medium

A particle is moving unidirectionally on a horizontal plane under the action of a constant power supplying energy source. The displacement (s) - time (t) graph that describes the motion of the particle is (graphs are drawn schematically and are not to scale) :

Options:

125

Medium

A bead of mass m stays at point P(a, b) on a wire bent in the shape of a parabola y = 4Cx2 and rotating with angular speed $\omega (see figure). The value of \omega $ is (neglect friction) :

Options:

A) 2\sqrt {2gC}

B) 2\sqrt {gC}

C) \sqrt {{{2gC} \over {ab}}}

D) \sqrt {{{2g} \over C}}

126

Medium

Consider a force $\overrightarrow F = - x\widehat i + y\widehat j$ . The work done by this force in moving a particle from point A(1, 0) to B(0, 1) along the line segment is : (all quantities are in SI units)

Options:

A) 2

B) {1 \over 2}

C) 1

D) {3 \over 2}

127

Medium

A particle of mass m is fixed to one end of a light spring having force constant k and unstretched length $\ell . The other end is fixed. The system is given an angular speed \omega $ about the fixed end of the spring such that it rotates in a circle in gravity free space. Then the stretch in the spring is :

Options:

A) {{m\ell {\omega ^2}} \over {k - m{\omega ^2}}}

B) {{m\ell {\omega ^2}} \over {k - m{\omega}}}

C) {{m\ell {\omega ^2}} \over {k + m{\omega ^2}}}

D) {{m\ell {\omega ^2}} \over {k + m{\omega}}}

128

Medium

An elevator in a building can carry a maximum of 10 persons, with the average mass of each person being 68 kg, The mass of the elevator itself is 920 kg and it moves with a constant speed of 3 m/s. The frictional force opposing the motion is 6000 N. If the elevator is moving up with its full capacity, the power delivered by the motor to the elevator (g = 10 m/s2) must be at least :

Options:

A) 48000 W

B) 62360 W

C) 56300 W

D) 66000 W

129

Medium

A smooth wire of length 2$\pi r is bent into a circle and kept in a vertical plane. A bead can slide smoothly on the wire. When the circle is rotating with angular speed \omega about the vertical diameter AB, as shown in figure, the bead is at rest with respect to the circular ring at position P as shown. Then the value of \omega $2 is equal to -

Options:

A) {{\sqrt 3 g} \over {2r}}

B) {{2g} \over {\left( {r\sqrt 3 } \right)}}

C) {{\left( {g\sqrt 3 } \right)} \over r}

D) {{2g} \over r}

130

Medium

A 60 HP electric motor lifts an elevator having a maximum total load capacity of 2000 kg. If the frictional force on the elevator is 4000 N, the speed of the elevator at full load is close to : (1 HP = 746 W, g = 10 ms-2)

Options:

A) 1.5 ms-1

B) 1.7 ms-1

C) 2.0 ms-1

D) 1.9 ms-1

131

Medium

A particle is moving along a circular path with a constant speed of 10 ms–1. What is the magnitude of the change in velocity of the particle, when it moves through an angle of 60o around the centre of the circle?

Options:

A) zero

B) 10 m/s

C) 10\sqrt 2 m/s

D) 10\sqrt 3 m/s

132

Medium

A uniform cable of mass 'M' and length 'L' is placed on a horizontal surface such that its (1/n)th part is hanging below the edge of the surface. To lift the hanging part of the cable upto the surface, the work done should be :

Options:

A) {{2MgL} \over {{n^2}}}

B) nMgL

C) {{MgL} \over {2{n^2}}}

D) {{MgL} \over {{n^2}}}

133

Medium

A body is projected at t = 0 with a velocity 10 ms–1 at an angle of 60o with the horizontal. The radius of curvature of its trajectory at t = 1s is R. neglecting air resistance and taking acceleration due to gravity g = 10 ms–2, the value of R is :

Options:

A) 2.8 m

B) 5.1 m

C) 2.5 m

D) 10.3 m

134

Medium

A particle moves in one dimension from rest under the influence of a force that varies with the distance travelled by the particle as shown in the figure. The kinetic energy of the particle after it has travelled 3m is :

Options:

A) 6.5 J

B) 2.5 J

C) 5 J

D) 4 J

135

Medium

A disc rotates about its axis of symmetry in a horizontal plane at a steady rate of $3.5$ revolutions per second. A coin placed at a distnce of 1.25 cm from the axis of rotation remains at rest on the disc. The coefficient of friction between the coin and the disc is : (g = 10 m/s2)

Options:

A) 0.5

B) 0.3

C) 0.7

D) 0.6

136

Medium

A particle which is experiencing a force, given by $\overrightarrow F = 3\widehat i - 12\widehat j, undergoes a displacement of \overrightarrow d = 4\overrightarrow i $ particle had a kinetic energy of 3 J at the beginning of the displacement, what is its kinetic energy at the end of the displacement ?

Options:

A) 9 J

B) 10 J

C) 12 J

D) 15 J

137

Medium

A conical pendulum of length 1 m makes an angle $\theta $ = 45o w.r.t. Z-axis and moves in a circle in the XY plane. The radius of the circle is 0.4 m and its center is vertically below O. The speed of the pendulum, in its circular path, will be: (Take g = 10 ms−2 )

Options:

A) 0.4 m/s

B) 4 m/s

C) 0.2 m/s

D) 2 m/s

138

Medium

A block of mass m is kept on a platform which starts from rest with constant acceleration g/2 upward, as shown in figure. Work done by normal reaction on block in time is -

Options:

A) {{m{g^2}{t^2}} \over 8}

B) {{3m{g^2}{t^2}} \over 8}

C) - {{m{g^2}{t^2}} \over 8}

D) 0

139

Medium

Two cars of masses m1 and m2 are moving in circles of radii r1 and r2, respectively. Their speeds are such that they make complete circles in the same time t. The ratio of their centripetal acceleration is

Options:

A) m1r1 : m2r2

B) m1 : m2

C) r1 : r2

D) 1 : 1

140

Medium

A force acts on a 2 kg object so that its position is given as a function of time as x = 3t2 + 5. What is the work done by this force in first 5 seconds ?

Options:

A) 850 J

B) 950 J

C) 875 J

D) 900 J

141

Medium

For a particle in uniform circular motion the acceleration $\overrightarrow a $ at a point P(R, θ) on the circle of radius R is (here θ is measured from the x–axis)

Options:

A) - {{{v^2}} \over R}\cos \theta \widehat i + {{{v^2}} \over R}\sin \theta \widehat j

B) - {{{v^2}} \over R}\sin \theta \widehat i + {{{v^2}} \over R}\cos \theta \widehat j

C) - {{{v^2}} \over R}\cos \theta \widehat i - {{{v^2}} \over R}\sin \theta \widehat j

D) {{{v^2}} \over R}\widehat i + {{{v^2}} \over R}\widehat j

142

Medium

A block of mass m, lying on a smooth horizontal surface, is attached to a sring (of negligible mass) of spring constant k. The other end of the spring is fixed, as shown in the figure. The block is initially at rest in its equilibrium position. If now the block is pulled with a constant force F, the maximum speed of the block is :

Options:

A) {{2F} \over {\sqrt {mk} }}

B) {F \over {\pi \sqrt {mk} }}

C) {{\pi F} \over {\sqrt {mk} }}

D) {F \over {\sqrt {mk} }}

143

Medium

A point $P moves in counter-clockwise direction on a circular path as shown in the figure. The movement of P is such that it sweeps out a length s = {t^3} + 5, where s is in metres and t is in seconds. The radius of the path is 20 m. The acceleration of 'P' when t=2 s$ is nearly.

Options:

A) 13m/{s_2}

B) 12m/{s^2}

C) 7.2m{s^2}

D) 14m/{s^2}

144

Medium

A body of mass m starts moving from rest along x-axis so that its velocity varies as $\upsilon = a\sqrt s $ where a is a constant and s is the distance covered by the body. The total work done by all the forces acting on the body in the first t seconds after the start of the motion is :

Options:

A) {1 \over 8}\,$ m a4 t2

B) 8 m a4 t2

C) 4 m a4 t2

D) {1 \over 4}\,$ m a4 t2

145

Medium

Which of the following statements is FALSE for a particle moving in a circle with a constant angular speed?

Options:

A) The velocity vector is tangent to the circle.

B) The acceleration vector is tangent to the circle.

C) The acceleration vector points to the centre of the circle.

D) The velocity and acceleration vectors are perpendicular to each other.

146

Medium

Two particles of the same mass m are moving in circular orbits because of force, given by $F\left( r \right) = {{ - 16} \over r} - {r^3}$ The first particle is at a distance r = 1, and the second, at r = 4. The best estimate for the ratio of kinetic energies of the first and the second particle is closest to :

Options:

A) 6 \times {10^{ - 2}}

B) 3 \times {10^{ - 3}}

C) {10^{ - 1}}

D) 6 \times {10^{ 2}}

147

Medium

The minimum velocity (in $m{s^{ - 1}}) with which a car driver must traverse a flat curve of radius 150 m and coefficient of friction 0.6$ to avoid skidding is

Options:

A) 60

B) 30

C) 15

D) 25

148

Medium

A particle is moving in a circular path of radius $a under the action of an attractive potential U = - {k \over {2{r^2}}}$ Its total energy is:

Options:

A) - {3 \over 2}{k \over {{a^2}}}

B) Zero

C) - {k \over {4{a^2}}}

D) {k \over {2{a^2}}}

149

Easy

A particle of charge 1.6 \mu \mathrm{C} and mass 16 \mu \mathrm{~g} is present in a strong magnetic field of 6.28 T . The particle is then fired perpendicular to magnetic field. The time required for the particle to return to original location for the first time is _________ s. (\pi=3.14)

Options:

150

Medium

An object is dropped from a height h from the ground. Every time it hits the ground it looses 50% of its kinetic energy. The total distance covered as t $ \to \infty $ is :

Options:

A) 3h

B) \infty

C) {5 \over 3}$h

D) {8 \over 3}$h

151

Medium

A string of length L is fixed at one end and carries a mass of M at the other end. The mass makes \left(\frac{3}{\pi}\right) rotations per second about the vertical axis passing through end of the string as shown. The tension in the string is __________ ML.

Options:

152

Medium

A body of mass m = 10–2 kg is moving in a medium and experiences a frictional force F = –kv2. Its initial speed is v0 = 10 ms–1. If, after 10 s, its energy is ${1 \over 8}mv_0^2$, the value of k will be:

Options:

A) 10-1 kg m-1 s-1

B) 10-3 kg m-1

C) 10-3 kg s-1

D) 10-4 kg m-1

153

Medium

A tube of length 1 m is filled completely with an ideal liquid of mass 2 M , and closed at both ends. The tube is rotated uniformly in horizontal plane about one of its ends. If the force exerted by the liquid at the other end is F then angular velocity of the tube is \sqrt{\frac{\mathrm{F}}{\alpha \mathrm{M}}} in SI unit. The value of \alpha is _________.

Options:

154

Medium

A time dependent force F = 6t acts on a particle of mass 1 kg. If the particle starts from rest, the work done by the force during the first 1 sec. will be:

Options:

A) 18 J

B) 4.5 J

C) 22 J

D) 9 J

155

Hard

A particle is moving in a circle of radius $50 \mathrm{~cm} in such a way that at any instant the normal and tangential components of it's acceleration are equal. If its speed at \mathrm{t}=0 is 4 \mathrm{~m} / \mathrm{s}, the time taken to complete the first revolution will be \frac{1}{\alpha}\left[1-e^{-2 \pi}\right] \mathrm{s}, where \alpha=$ _________.

Options:

156

Medium

Velocity-time graph for a body of mass 10 kg is shown in figure. Work-done on the body in first two seconds of the motion is :

Options:

A) 12000 J

B) -$ 12000 J

C) -$ 4500 J

D) -$ 9300 J

157

Easy

A stone tied to 180 \mathrm{~cm} long string at its end is making 28 revolutions in horizontal circle in every minute. The magnitude of acceleration of stone is \frac{1936}{x} ms^{-2}. The value of x ________. (Take \pi=\frac{22}{7} )

Options:

158

Medium

A particle of mass M is moving in a circle of fixed radius R in such a way that its centripetal acceleration at time t is given by n2 R t2 where n is a constant. The power delivered to the particle by the force acting on it, is :

Options:

A) M n2 R2 t

B) M n R2 t

C) M n R2 t2

D) {1 \over 2}$ M n2 R2 t2

159

Hard

A car is moving on a circular path of radius 600 m such that the magnitudes of the tangential acceleration and centripetal acceleration are equal. The time taken by the car to complete first quarter of revolution, if it is moving with an initial speed of 54 km/hr is $t(1-e^{-\pi/2})s$. The value of t is ____________.

Options:

160

Medium

A car of weight W is on an inclined road that rises by 100 m over a distance of 1 km and applies a constant frictional force ${W \over 20} on the car. While moving uphill on the road at a speed of 10 ms−1, the car needs power P. If it needs power {p \over 2} while moving downhill at speed v then value of \upsilon $ is :

Options:

A) 20 ms$-$1

B) 15 ms$-$1

C) 10 ms$-$1

D) 5 ms$-$1

161

Easy

A person starts his journey from centre 'O' of the park and comes back to the same position following path OPQO as shown in the figure. The radius of path taken by the person is 200 m and he takes 3 min 58 sec to complete his journey. The average speed of the person is _____________ ms$-1. (take \pi$ = 3.14)

Options:

162

Medium

A point particle of mass $m, moves long the uniformly rough track PQR as shown in the figure. The coefficient of friction, between the particle and the rough track equals \mu . The particle is released, from rest from the point P and it comes to rest at point R. The energies, lost by the ball, over the parts, PQ and QR, of the track, are equal to each other , and no energy is lost when particle changes direction from PQ to QR. The value of the coefficient of friction \mu and the distance x (=QR),$ are, respectively close to:

Options:

A) 0.29 and 3.5 m

B) 0.29 and 6.5 m

C) 0.2 and 6.5 m

D) 0.2 and 3.5 m

163

Hard

A pendulum of length 2 m consists of a wooden bob of mass 50 g. A bullet of mass 75 g is fired towards the stationary bob with a speed v. The bullet emerges out of the bob with a speed ${v \over 3} and the bob just completes the vertical circle. The value of v is ___________ ms-$1. (if g = 10 m/s2).

Options:

164

Medium

A person trying to lose weight by burning fat lifts a mass of $10 kg upto a height of 1 m 1000 times. Assume that the potential energy lost each time he lowers the mass is dissipated. How much fat will he use up considering the work done only when the weight is lifted up? Fat supplies 3.8 \times {10^7}J of energy per kg which is converted to mechanical energy with a 20\% efficiency rate. Take g = 9.8\,m{s^{ - 2}}$ :

Options:

A) 9.89 \times {10^{ - 3}}\,\,kg

B) 12.89 \times {10^{ - 3}}\,kg

C) 2.45 \times {10^{ - 3}}\,\,kg

D) 6.45 \times {10^{ - 3}}\,\,kg

165

Medium

A curved in a level road has a radius 75 m. The maximum speed of a car turning this curved road can be 30 m/s without skidding. If radius of curved road is changed to 48 m and the coefficient of friction between the tyres and the road remains same, then maximum allowed speed would be ___________ m/s.

Options:

166

Medium

When a rubber-band is stretched by a distance $x, it exerts restoring force of magnitude F = ax + b{x^2} where a and b are constants. The work done in stretching the unstretched rubber-band by L$ is :

Options:

A) a{L^2} + b{L^3}

B) {1 \over 2}\left( {a{L^2} + b{L^3}} \right)

C) {{a{L^2}} \over 2} + {{b{L^3}} \over 3}

D) {1 \over 2}\left( {{{a{L^2}} \over 2} + {{b{L^3}} \over 3}} \right)

167

Medium

A small bob tied at one end of a thin string of length 1 m is describing a vertical circle so that the maximum and minimum tension in the string are in the ratio 5 : 1. The velocity of the bob at the highest position is ________ m/s. (Take g = 10 m/s2)

Options:

168

Medium

This question has Statement $1 and Statement 2. Of the four choices given after the Statements, choose the one that best describes the two Statements. If two springs {S_1} and {S_2} of force constants {k_1} and {k_2}, respectively, are stretched by the same force, it is found that more work is done on spring {S_1} than on spring {S_2}. STATEMENT 1: If stretched by the same amount work done on {S_1}, Work done on {S_1} is more than {S_2} STATEMENT 2: {k_1} < {k_2}

Options:

A) Statement 1 is false, Statement 2 is true

B) Statement 1 is true, Statement 2 is false

C) Statement 1 is true, Statement 2 is true, Statement 2 is the correct explanation for Statement 1

D) Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for Statement 1

169

Medium

A stone of mass ' m ' kg is tied to a string of length ' L ' m and moved in a vertical circle of radius 49 cm in a vertical plane. If it completes 30 revolutions per minute, the tension in the string when it is at the lowermost point is nearly [Take \pi^2=10 and acceleration due to gravity, \mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2 ]

Options:

A) (90 \mathrm{~m}) \mathrm{N}

B) \quad(60 \mathrm{~m}) \mathrm{N}

C) (45 m) \mathrm{N}

D) (15 m) \mathrm{N}

170

Medium

The potential energy function for the force between two atoms in a diatomic molecule is approximately given by $U\left( x \right) = {a \over {{x^{12}}}} - {b \over {{x^6}}}, where a and b are constants and x is the distance between the atoms. If the dissociation energy of the molecule is D = \left[ {U\left( {x = \infty } \right) - {U_{at\,\,equilibrium}}} \right],\,\,D$ is

Options:

A) {{{b^2}} \over {2a}}

B) {{{b^2}} \over {12a}}

C) {{{b^2}} \over {4a}}

D) {{{b^2}} \over {6a}}

171

Medium

A point mass ' m ' attached at one end of a massless, inextensible string of length ' l ' performs a vertical circular motion and the string rotates in vertical plane, as shown in the diagram. The increase in the centripetal acceleration of the point mass when it moves from point A to point C is [ \mathrm{g}= acceleration due to gravity.]

Options:

A) 3 g

B) 2 g

C) g

D) \frac{\mathrm{g}}{2}

172

Medium

An athlete in the olympic games covers a distance of $100 m in 10 s.$ His kinetic energy can be estimated to be in the range

Options:

A) 200J-500J

B) 2 \times {10^5}J - 3 \times {10^5}J

C) 20,000J - 50,000J

D) 2,000J - 5,000J

173

Medium

An inextensible string of length ' l ' fixed at one end, carries a mass ' m ' at the other end. If the string makes \frac{1}{\pi} revolutions per second around the vertical axis through the fixed end, the tension in the string is [The string makes an angle \theta with the vertical]

Options:

A) 16 ml

B) 8 ml

C) 4 ml

D) 2 ml

174

Medium

A $2 kg block slides on a horizontal floor with a speed of 4m/s. It strikes a uncompressed spring, and compress it till the block is motionless. The kinetic friction force is 15N and spring constant is 10, 000 N/m.$ The spring compresses by

Options:

A) 8.5cm

B) 5.5cm

C) 2.5cm

D) 11.0cm

175

Medium

A particle describes a horizontal circle on smooth inner surface of a cone as shown in figure. If the height of the circle above the vertex is 10 cm . The speed of the particle is \left(\mathrm{g}\right., acceleration due to gravity \left.=10 \mathrm{~m} / \mathrm{s}^2\right)

Options:

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

B) 1.5 \mathrm{~m} / \mathrm{s}

C) 1 \mathrm{~m} / \mathrm{s}

D) 0.5 \mathrm{~m} / \mathrm{s}

176

Medium

A particle is projected at $60^\circ $ to the horizontal with a kinetic energy K. The kinetic energy at the highest point is

Options:

A) K/2

B) K

C) Zero

D) K/4

177

Medium

Two stones of masses m and 3 m are whirled in horizontal circles, the heavier one in a radius \left(\frac{\mathrm{r}}{3}\right) and lighter one in a radius r . The tangential speed of lighter stone is ' n ' times the value of heavier stone. When the magnitude of centripetal force becomes equal the value of n is

Options:

A) 4

B) 3

C) 2

D) 1

178

Medium

A ball of mass $0.2 kg is thrown vertically upwards by applying a force by hand. If the hand moves 0.2 m while applying the force and the ball goes upto 2 m height further, find the magnitude of the force. (consider g = 10\,m/{s^2}$).

Options:

A) 4N

B) 16 N

C) 20 N

D) 22 N

179

Medium

A motor cyclist has to rotate in horizontal circles inside the cylindrical wall of inner radius ' R ' metre. If the coefficient of friction between the wall and the tyres is ' \mu_{\mathrm{s}} ', then the minimum speed required is ( \mathrm{g}= acceleration due to gravity)

Options:

A) \sqrt{\mu_{\mathrm{r}} \mathrm{Rg}}

B) \sqrt{\frac{\mathrm{Rg}}{\mu_{\mathrm{s}}}}

C) \sqrt{\frac{\mu_{\mathrm{s}}}{\mathrm{Rg}}}

D) \sqrt{\frac{R^2 g}{\mu_s}}

180

Medium

A mass of $M kg is suspended by a weightless string. The horizontal force that is required to displace it until the string makes an angle of {45^ \circ }$ with the initial vertical direction is

Options:

A) Mg\left( {\sqrt 2 + 1} \right)

B) Mg\sqrt 2

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

D) Mg\left( {\sqrt 2 - 1} \right)

181

Medium

The figure shows two masses ' m ' and ' M ' connected by a light string that passes through { }_a small hole ' O ' at the centre of the table. Mass ' m ' is moved round in a horizontal circle with ' O ' as the centre. The frequency with which ' m ' should be revolved so that ' M ' remains stationary is ( \mathrm{g}= gravitational acceleration)

Options:

A) \frac{1}{\pi} \sqrt{\frac{\mathrm{ML}}{\mathrm{mg}}}

B) \frac{1}{2 \pi} \sqrt{\frac{\mathrm{Mg}}{\mathrm{mL}}}

C) \frac{1}{\pi} \sqrt{\frac{\mathrm{Mg}}{\mathrm{mL}}}

D) \frac{1}{2 \pi} \sqrt{\frac{\mathrm{ML}}{\mathrm{mg}}}

182

Medium

A particle of mass $100g is thrown vertically upwards with a speed of 5 m/s$. The work done by the force of gravity during the time the particle goes up is

Options:

A) -0.5J

B) -1.25J

C) 1.25J

D) 0.5J

183

Medium

Radius of curved road is ' R ', width of road is ' b '. The outer edge of road is raised by ' h ' with respect to inner edge so that a car with velocity ' V ' can pass safe over it, then value of ' h ' is ( \mathrm{g}= acceleration due to gravity)

Options:

A) \frac{\mathrm{V}^2 \mathrm{~b}}{\mathrm{Rg}}

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

C) \frac{V^2 R}{g}

D) \frac{V^2 b}{g}

184

Medium

The potential energy of a $1 kg particle free to move along the x-axis is given by V\left( x \right) = \left( {{{{x^4}} \over 4} - {{{x^2}} \over 2}} \right)J. The total mechanical energy of the particle is 2J. Then, the maximum speed (in m/s$) is

Options:

A) {3 \over {\sqrt 2 }}

B) {\sqrt 2 }

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

D) 2

185

Medium

Two bodies of mass 10 kg and 5 kg are moving in concentric circular orbits of radii ' R ' and ' r ' respectively such that their periods are same. The ratio between their centripetal acceleration is

Options:

A) \mathrm{R} / \mathrm{r}

B) \quad \mathrm{r} / \mathrm{R}

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

D) r^2 / R^2

186

Medium

A body of mass $m is accelerated uniformly from rest to a speed v in a time T.$ The instantaneous power delivered to the body as a function of time is given by

Options:

A) {{m{v^2}} \over {{T^2}}}.{t^2}

B) {{m{v^2}} \over {{T^2}}}.t

C) {1 \over 2}{{m{v^2}} \over {{T^2}}}.{t^2}

D) {1 \over 2}{{m{v^2}} \over {{T^2}}}.t

187

Medium

A car is driven on the banked road of radius of curvature 20 m with maximum safe speed. In order to increase its safety speed by 20 \%, without changing the angle of banking, the increase in the radius of curvature will be [Assume friction is same on the road]

Options:

A) 28.8 m

B) 14.4 m

C) 8.8 m

D) 4.8 m

188

Medium

A bullet fired into a fixed target loses half of its velocity after penetrating $3 cm.$ How much further it will penetrate before coming to rest assuming that it faces constant resistance to motion?

Options:

A) 2.0 cm

B) 3.0 cm

C) 1.0 cm

D) 1.5 cm

189

Medium

A vehicle is moving with uniform speed along 3 different shaped roads as horizontal, concave and convex. The surface of road on which, the normal reaction on vehicle is maximum is

Options:

A) convex

B) concave

C) horizontal

D) same on all the 3 surface

190

Medium

A spherical ball of mass $20 kg is stationary at the top of a hill of height 100 m. It rolls down a smooth surface to the ground, then climbs up another hill of height 30 m and finally rolls down to a horizontal base at a height of 20 m$ above the ground. The velocity attained by the ball is

Options:

A) 20 m/s

B) 40 m/s

C) 10\sqrt {30} \,\,\,m/s

D) 10\,\,m/s

191

Medium

A vehicle is moving with a constant speed of 10 \mathrm{~m} / \mathrm{s} in a circular horizontal track of radius 20 m . A bob is suspended from the roof of a vehicle by a massless string. The angle made by the string with the vertical will be (acceleration due to gravity, \mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2 )

Options:

A) \tan ^{-1}(0.5)

B) \tan ^{-1}(0.6)

C) \tan ^{-1}(0.7)

D) \tan ^{-1}(0.8)

192

Medium

The upper half of an inclined plane with inclination $\phi $ is perfectly smooth while the lower half is rough. A body starting from rest at the top will again come to rest at the bottom if the coefficient of friction for the lower half is given by

Options:

A) 2\,\cos \,\,\phi

B) 2\,sin\,\,\phi

C) \,\tan \,\,\phi

D) 2\,\tan \,\,\phi

193

Medium

A body of mass 100 gram is tied to a spring of spring constant 8 \mathrm{~N} / \mathrm{m}, while the other end of a spring is fixed. If the body moves in a circular path on smooth horizontal surface with constant angular speed 8 \mathrm{rad} / \mathrm{s} then the ratio of extension in the spring to its natured length will be

Options:

A) 1: 1

B) 8: 1

C) 2: 1

D) 4: 1

194

Medium

A particle moves in a straight line with retardation proportional to its displacement. Its loss of kinetic energy for any displacement $x$ is proportional to

Options:

A) x

B) {e^x}

C) {x^2}

D) {\log _e}x

195

Medium

A simple pendulum oscillates with an angular amplitude \theta. If the maximum tension in the string is 4 times the minimum tension then the value of \theta is

Options:

A) \cos ^{-1}(0.75)

B) \cos ^{-1}(0.5)

C) \sin ^{-1}(0.5)

D) \sin ^{-1}(0.75)

196

Medium

A uniform chain of length $2 m is kept on a table such that a length of 60 cm hangs freely from the edge of the table. The total mass of the chain is 4 kg.$ What is the work done in pulling the entire chain on the table?

Options:

A) 12 J

B) 3.6 J

C) 7.2 J

D) 1200 J

197

Medium

A pendulum bob has a speed 4 \mathrm{~m} / \mathrm{s} at its lowest position. The pendulum is 1 m long. When the length of the string makes an angle of 60^{\circ} with the vertical, the speed of the bob at that position is (acceleration due to gravity, \mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2, \cos 60^{\circ}=0.5 )

Options:

A) 6 \mathrm{~m} / \mathrm{s}

B) \sqrt{3} \mathrm{~m} / \mathrm{s}

C) \sqrt{6} \mathrm{~m} / \mathrm{s}

D) 3 \mathrm{~m} / \mathrm{s}

198

Medium

A particle is acted upon by a force of constant magnitude which is always perpendicular to the velocity of the particle, the motion of the particles takes place in a plane. It follows that

Options:

A) its kinetic energy is constant

B) is acceleration is constant

C) its velocity is constant

D) it moves in a straight line

199

Medium

A wheel initially at rest, begins to rotate about its axis with constant angular acceleration. If it rotates through an angle \theta_1 in first 2 s and a further angle \theta_2 in the next 2 s , the ratio \theta_1: \theta_2 is

Options:

A) 1: 6

B) 6: 1

C) 3: 1

D) 1: 3

200

Medium

A body of mass $' m ', acceleration uniformly from rest to '{v_1}' in time {T}$. The instantaneous power delivered to the body as a function of time is given by

Options:

A) {{m{v_1}{t^2}} \over {{T}}}

B) {{mv_1^2t} \over {T^2}}

C) {{m{v_1}t} \over {{T}}}

D) {{mv_1^2t} \over {{T}}}

201

Medium

For a particle moving in a circle with constant angular speed, which of the following statements is 'false'?

Options:

A) The velocity vector is tangent to the circle.

B) The acceleration vector is tangent to the circle.

C) The velocity and acceleration vectors are perpendicular to each other.

D) The acceleration vector points to the centre of the circle.

202

Medium

A force $\overrightarrow F = \left( {5\overrightarrow i + 3\overrightarrow j + 2\overrightarrow k } \right)N is applied over a particle which displaces it from its origin to the point \overrightarrow r = \left( {2\overrightarrow i - \overrightarrow j } \right)m.$ The work done on the particle in joules is

Options:

A) +10

B) +7

C) -7

D) +13

203

Medium

A particle performing uniform circular motion of radius \frac{\pi}{2} \mathrm{~m} makes x revolutions in time t. Its tangential velocity is

Options:

A) \frac{x}{\pi t}

B) \frac{\pi^2}{x t}

C) \frac{\pi^2 x}{t}

D) \frac{\pi x}{t}

204

Medium

A wire suspended vertically from one of its ends is stretched by attaching a weight of $200N to the lower end. The weight stretches the wire by 1 mm.$ Then the elastic energy stored in the wire is

Options:

A) 0.2 J

B) 10 J

C) 20 J

D) 0.1 J

205

Medium

A weightless thread can bear tension up to 3.7 kg wt. A stone of mass 500 gram is tied to it and revolved in circular path of radius 4 m in vertical plane. Maximum angular velocity of the stone will be (acceleration due to gravity, \mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2 )

Options:

A) 16 \mathrm{rad} / \mathrm{s}

B) 4 \mathrm{rad} / \mathrm{s}

C) 2 \mathrm{rad} / \mathrm{s}

D) 8 \mathrm{rad} / \mathrm{s}

206

Medium

A spring of spring constant $5 \times {10^3}\,N/m is stretched initially by 5 cm from the unstretched position. Then the work required to stretch it further by another 5 cm$ is

Options:

A) 12.50 N-m

B) 18.75 N-m

C) 25.00 N-m

D) 625 N-m

207

Medium

When a ceiling fan is switched off, its angular velocity falls to \left(\frac{1}{3}\right)^{\text {rd }} while it makes 24 rotations. How many more rotations will it make before coming to rest?

Options:

A) 3

B) 6

C) 9

D) 12

208

Medium

A body is moved along a straight line by a machine delivering a constant power. The distance moved by the body in time $'t'$ is proportional to

Options:

A) {t^{3/4}}

B) {t^{3/2}}

C) {t^{1/4}}

D) {t^{1/2}}

209

Medium

The linear speed of a particle at the equator of the earth due to its spin motion is ' V '. The linear speed of the particle at latitude 30^{\circ} is $\left[\begin{array}{l} \sin 30^{\circ}=\cos 60^{\circ}=1 / 2 \\ \cos 30^{\circ}=\sin 60^{\circ}=\sqrt{3} / 2 \end{array}\right]

Options:

A) \frac{\mathrm{V}}{\sqrt{2}}

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

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

D) \mathrm{V}

210

Medium

A spring of force constant $800 N/m has an extension of 5 cm. The work done in extending it from 5 cm to 15 cm$ is

Options:

A) 16J

B) 8J

C) 32J

D) 24J

211

Medium

Two objects of masses ' m_1 ' and ' m_2 ' are moving in the circles of radii ' r_1 ' and ' r_2 ' respectively. Their respective angular speeds ' \omega_1 ' and ' \omega_2 ' are such that they both complete one revolution in the same time ' t '. The ratio of linear speed of ' m_2 ' to that of ' m_1 ' is

Options:

A) \omega_1: \omega_2

B) \mathrm{T}_2: \mathrm{T}_1

C) \mathrm{m}_1: \mathrm{m}_2

D) \mathrm{r_2: r_1}

212

Medium

A ball whose kinetic energy E, is projected at an angle of $45^\circ $ to the horizontal. The kinetic energy of the ball at the highest point of its height will be

Options:

A) E

B) {E \over {\sqrt 2 }}

C) {E \over 2}

D) zero

213

Medium

A body performing uniform circular motion of radius ' R ' has frequency ' n '. Its centripetal acceleration per unit radius is proportional to (n)^x. The value of x is

Options:

A) 1

B) 2

C) -1

D) -2

214

Medium

If a body looses half of its velocity on penetrating $3 cm$ in a wooden block, then how much will it penetrate more before coming to rest?

Options:

A) 1 cm

B) 2 cm

C) 3 cm

D) 4 cm

215

Medium

A particle starting from rest moves along the circumference of a circle of radius ' r ' with angular acceleration ' \alpha '. The magnitude of the average velocity in time it completes the small angular displacement ' \theta ' is

Options:

A) \frac{r^2}{2 \alpha \theta}

B) \frac{\mathrm{r}}{2 \alpha \theta}

C) \frac{\mathrm{r} \alpha \theta}{2}

D) \frac{\mathrm{r}}{\sqrt{2}} \sqrt{\alpha \theta}

216

Easy

In a hydraulic lift, the surface area of the input piston is 6 cm2 and that of the output piston is 1500 cm2. If 100 N force is applied to the input piston to raise the output piston by 20 cm, then the work done is _______ kJ.

Options:

217

Medium

A particle is moving in a circle with uniform speed. It has constant

Options:

A) velocity.

B) acceleration.

C) kinetic energy.

D) displacement.

218

Medium

A force \mathrm{f}=\mathrm{x}^2 \mathrm{y} \hat{\mathrm{i}}+\mathrm{y}^2 \hat{\mathrm{j}} acts on a particle in a plane \mathrm{x}+\mathrm{y}=10. The work done by this force during a displacement from (0,0) to (4 \mathrm{~m}, 2 \mathrm{~m}) is _________ Joule (round off to the nearest integer)

Options:

219

Medium

A particle of mass ' m ' is performing uniform circular motion along a circular path of radius ' r '. Its angular momentum about the axis passing through the centre and perpendicular to the plane is ' L '. The kinetic energy of the particle is

Options:

A) \frac{L^2}{2 Mr^2}

B) \frac{2 \mathrm{~L}^2}{\mathrm{mr}^2}

C) \frac{\mathrm{L}^2}{\mathrm{mr}^2}

D) \frac{2 \mathrm{~L}^2}{3 \mathrm{mr}^2}

220

Easy

A force $(3 x^2+2 x-5) \mathrm{N} displaces a body from x=2 \mathrm{~m} to x=4 \mathrm{~m}$. Work done by this force is ________ J.

Options:

221

Medium

A particle of mass ' m ' performs uniform circular motion of radius ' r ' with linear speed ' v ' under the application of force ' F '. If ' m ', ' v ' and ' \mathrm{r} ' are all increased by 20 \% the necessary change in force required to maintain the particle in uniform circular motion, is

Options:

A) 12 \%

B) 44 \%

C) 14 \%

D) 144 \%

222

Easy

A block of mass 10 \mathrm{~kg} is moving along \mathrm{x}-axis under the action of force F=5 x~ N. The work done by the force in moving the block from x=2 m to 4 m will be __________ J.

Options:

223

Medium

A particle rotates in a horizontal circle of radius 'R' in a conical funnel with constant speed 'V'. The inner surface of the funnel is smooth. The height of the plane of the circle from the vertex of the funnel is (g-acceleration due to gravity)

Options:

A) \frac{V}{g}

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

C) \frac{\mathrm{V}^2}{2 \mathrm{~g}}

D) \frac{\mathrm{V}^2}{\mathrm{~g}}

224

Easy

A car accelerates from rest to $u \mathrm{~m} / \mathrm{s}. The energy spent in this process is E J. The energy required to accelerate the car from u \mathrm{~m} / \mathrm{s} to 2 \mathrm{u} \mathrm{m} / \mathrm{s} is \mathrm{nE~J}. The value of \mathrm{n}$ is ____________.

Options:

225

Medium

For a particle in uniform circular motion

Options:

A) linear velocity always radial to the circular path, without change in its magnitude

B) linear velocity always tangential to the circular path, without change in its magnitude

C) linear acceleration always tangential to the circular path

D) linear acceleration always along the axis of the circular path

226

Easy

To maintain a speed of 80 km/h by a bus of mass 500 kg on a plane rough road for 4 km distance, the work done by the engine of the bus will be ____________ KJ. [The coefficient of friction between tyre of bus and road is 0.04.]

Options:

227

Medium

A disc at rest is subjected to a uniform angular acceleration about its axis. Let \theta and \theta_1 be the angle made by the disc in 2^{\text {nd }} and 3^{\text {rd }} second of its motion. The ratio \frac{\theta}{\theta_1} is

Options:

A) 2: 3

B) 1: 2

C) 2: 3

D) 4: 5

228

Easy

A block of mass $5 \mathrm{~kg} starting from rest pulled up on a smooth incline plane making an angle of 30^{\circ} with horizontal with an affective acceleration of 1 \mathrm{~ms}^{-2}. The power delivered by the pulling force at t=10 \mathrm{~s} from the start is ___________ W. [use \mathrm{g}=10 \mathrm{~ms}^{-2}$ ] (calculate the nearest integer value)

Options:

229

Medium

A body moves along a circular path of radius 15 cm . It starts from a point on the circular path and reaches the end of diameter in 3 second, The angular speed of the body in \mathrm{rad} / \mathrm{s} is

Options:

A) \frac{\pi}{2}

B) \frac{\pi}{3}

C) \frac{\pi}{4}

D) \frac{\pi}{5}

230

Easy

A force $\vec{F}=(2+3 x) \hat{i} acts on a particle in the x direction where F is in newton and x is in meter. The work done by this force during a displacement from x=0 to x=4 \mathrm{~m}$, is __________ J.

Options:

231

Medium

A wheel of radius 1 m rolls through 180^{\circ} over a plane surface. The magnitude of the displacement of the point of the wheel initially in contact with the surface is.

Options:

A) 2 \pi

B) \pi

C) \sqrt{\pi^2+4}

D) 3 \pi

232

Easy

If the maximum load carried by an elevator is $1400 \mathrm{~kg} ( 600 \mathrm{~kg} - Passengers + 800 \mathrm{kg} - elevator), which is moving up with a uniform speed of 3 \mathrm{~m} \mathrm{~s}^{-1} and the frictional force acting on it is 2000 \mathrm{~N}, then the maximum power used by the motor is __________ \mathrm{kW}\left(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\right)

Options:

233

Medium

The string of pendulum of length ' L ' is displaced through 90^{\circ} from the vertical and released. Then the maximum strength of the string in order to withstand the tension, as the pendulum passes through the mean position is ( \mathrm{m}= mass of pendulum, \mathrm{g}= acceleration due to gravity)

Options:

A) mg

B) 3 mg

C) 5 mg

D) 6 mg

234

Medium

A closed circular tube of average radius 15 cm, whose inner walls are rough, is kept in vertical plane. A block of mass 1 kg just fit inside the tube. The speed of block is 22 m/s, when it is introduced at the top of tube. After completing five oscillations, the block stops at the bottom region of tube. The work done by the tube on the block is __________ J. (Given g = 10 m/s$^2$).

Options:

235

Medium

A particle at rest starts moving with a constant angular acceleration of 4 \mathrm{~rad} / \mathrm{s}^2 in a circular path. The time at which magnitudes of its centripetal acceleration and tangential acceleration will be equal, is (in second)

Options:

A) \frac{1}{4}

B) \frac{1}{3}

C) \frac{1}{2}

D) \frac{2}{3}

236

Easy

A body of mass $5 \mathrm{~kg} is moving with a momentum of 10 \mathrm{~kg} \mathrm{~ms}^{-1}. Now a force of 2 \mathrm{~N} acts on the body in the direction of its motion for 5 \mathrm{~s}. The increase in the Kinetic energy of the body is ___________ \mathrm{J}$.

Options:

237

Medium

A particle is performing uniform circular motion along the circumference of the circle of diameter 1 m with frequency 4 Hz . The acceleration of the particle in \mathrm{m} / \mathrm{s}^2 is

Options:

A) 8 \pi^2

B) 16 \pi^2

C) 24 \pi^2

D) 32 \pi^2

238

Easy

A body is dropped on ground from a height '$h_{1}' and after hitting the ground, it rebounds to a height 'h_{2}'. If the ratio of velocities of the body just before and after hitting ground is 4 , then percentage loss in kinetic energy of the body is \frac{x}{4}. The value of x$ is ____________.

Options:

239

Medium

A particle moves around a circular path of radius '$r' with uniform speed 'V$'. After moving half the circle, the average acceleration of the particle is

Options:

A) \frac{\mathrm{V}^2}{\mathrm{r}}

B) \frac{2 V^2}{r}

C) \frac{2 V^2}{\pi r}

D) \frac{\mathrm{V}^2}{\pi \mathrm{r}}

240

Medium

A particle of mass $10 \mathrm{~g} moves in a straight line with retardation 2 x, where x is the displacement in SI units. Its loss of kinetic energy for above displacement is \left(\frac{10}{x}\right)^{-n} J. The value of \mathrm{n}$ will be __________

Options:

241

Medium

On dry road, the maximum speed of a vehicle along a circular path is '$V'. When the road becomes wet, maximum speed becomes \frac{\mathrm{V}}{2}. If coefficient of friction of dry road is '\mu$' then that of wet road is

Options:

A) \frac{2 \mu}{3}

B) \frac{\mu}{4}

C) \frac{\mu}{3}

D) \frac{3 \mu}{4}

242

Easy

A block is fastened to a horizontal spring. The block is pulled to a distance $x=10 \mathrm{~cm} from its equilibrium position (at x=0) on a frictionless surface from rest. The energy of the block at x=5 \mathrm{cm} is 0.25 \mathrm{~J}. The spring constant of the spring is ___________ \mathrm{Nm}^{-1}

Options:

243

Medium

A string of length '$L' fixed at one end carries a body of mass '\mathrm{m}' at the other end. The mass is revolved in a circle in the horizontal plane about a vertical axis passing through the fixed end of the string. The string makes angle '\theta' with the vertical. The angular frequency of the body is '\omega$'. The tension in the string is

Options:

A) \mathrm{mL}^2 \omega

B) \mathrm{mL} \omega^2

C) \frac{\omega^2}{\mathrm{~mL}}

D) \frac{\mathrm{m} \omega^2}{\mathrm{~L}}

244

Easy

A force $\mathrm{F}=\left(5+3 y^{2}\right) acts on a particle in the y-direction, where \mathrm{F} is in newton and y is in meter. The work done by the force during a displacement from y=2 \mathrm{~m} to y=5 \mathrm{~m}$ is ___________ J.

Options:

245

Medium

A stone is projected at angle $\theta with velocity u. If it executes nearly a circular motion at its maximum point for short time, then the radius of the circular path will be ( g=$ acceleration due to gravity)

Options:

A) \frac{u^2}{g}

B) \frac{u^2 \cos ^2 \theta}{g}

C) \frac{u^2 \sin ^2 \theta}{g}

D) \frac{u^2 \cos ^2 \theta}{2 g}

246

Easy

A small particle moves to position $5 \hat{i}-2 \hat{j}+\hat{k} from its initial position 2 \hat{i}+3 \hat{j}-4 \hat{k} under the action of force 5 \hat{i}+2 \hat{j}+7 \hat{k} \mathrm{~N}$. The value of work done will be __________ J.

Options:

247

Medium

A particle is moving in a circle with uniform speed '$v$'. In moving from a point to another diametrically opposite point

Options:

A) the momentum changes by $\mathrm{mv}

B) the momentum changes by $2 \mathrm{~mv}

C) the kinetic energy changes by $\frac{1}{2} \mathrm{mv}^2

D) the kinetic energy changes by $\mathrm{m v^2}

248

Easy

A lift of mass $\mathrm{M}=500 \mathrm{~kg} is descending with speed of 2 \mathrm{~ms}^{-1}. Its supporting cable begins to slip thus allowing it to fall with a constant acceleration of 2 \mathrm{~ms}^{-2}. The kinetic energy of the lift at the end of fall through to a distance of 6 \mathrm{~m} will be _____________ \mathrm{kJ}$.

Options:

249

Medium

A body of mass '$\mathrm{m}' attached at the end of a string is just completing the loop in a vertical circle. The apparent weight of the body at the lowest point in its path is ( \mathrm{g}$ = gravitational acceleration)

Options:

A) zero

B) \mathrm{mg}

C) 3 \mathrm{~mg}

D) 6 \mathrm{~mg}

250

Medium

A body of mass 2 \mathrm{~kg} is initially at rest. It starts moving unidirectionally under the influence of a source of constant power P. Its displacement in 4 \mathrm{~s} is \frac{1}{3} \alpha^{2} \sqrt{P} m. The value of \alpha will be ______.

Options:

251

Medium

A railway track is banked for a speed ',$v' by elevating outer rail by a height 'h' above the inner rail. The distance between two rails is 'd' then the radius of curvature of track is ( \mathrm{g}=$ gravitational acceleration)

Options:

A) \frac{\mathrm{v}^2 \mathrm{~d}}{\mathrm{gh}}

B) \mathrm{\frac{2 v^2}{g d h}}

C) \mathrm{\frac{g d}{2 v^2 h}}

D) \mathrm{\frac{v^2}{2 g h d}}

252

Medium

A 0.4 kg mass takes 8s to reach ground when dropped from a certain height 'P' above surface of earth. The loss of potential energy in the last second of fall is __________ J. (Take g = 10 m/s$^2$)

Options:

253

Medium

Two particles having mass '$M' and 'm' are moving in a circular path with radius 'R' and 'r$' respectively. The time period for both the particles is same. The ratio of angular velocity of the first particle to the second particle will be

Options:

A) 1 : 1

B) 1 : 2

C) 2 : 3

D) 3 : 4

254

Medium

An object of mass 'm' initially at rest on a smooth horizontal plane starts moving under the action of force F = 2N. In the process of its linear motion, the angle $\theta (as shown in figure) between the direction of force and horizontal varies as \theta=\mathrm{k}x, where k is a constant and x is the distance covered by the object from its initial position. The expression of kinetic energy of the object will be E = {n \over k}\sin \theta $. The value of n is ___________.

Options:

255

Medium

In a conical pendulum the bob of mass '$\mathrm{m}' moves in a horizontal circle of radius 'r' with uniform speed '\mathrm{V}'. The string of length '\mathrm{L}' describes a cone of semi vertical angle '\theta'. The centripetal force acting on the bob is ( \mathrm{g}=$ acceleration due to gravity)

Options:

A) \frac{\mathrm{mgr}}{\sqrt{\mathrm{L}^2-\mathrm{r}^2}}

B) \frac{\mathrm{mgr}}{\left(\mathrm{L}^2-\mathrm{r}^2\right)}

C) \frac{\sqrt{\mathrm{L}^2-\mathrm{r}^2}}{\mathrm{mgL}}

D) \frac{\mathrm{mgL}}{\sqrt{\mathrm{L}^2-\mathrm{r}^2}}

256

Medium

A body of mass 1kg begins to move under the action of a time dependent force $\overrightarrow F = \left( {t\widehat i + 3{t^2}\,\widehat j} \right) N, where \widehat i and \widehat j are the unit vectors along x and y$ axis. The power developed by above force, at the time t = 2s, will be ____________ W.

Options:

257

Medium

A ball of mass '$\mathrm{m}' is attached to the free end of a string of length 'l'. The ball is moving in horizontal circular path about the vertical axis as shown in the diagram. The angular velocity '\omega' of the ball will be [ \mathrm{T}=$ Tension in the string.]

Options:

A) \sqrt{\frac{\mathrm{T} l}{\mathrm{~m}}}

B) \sqrt{\frac{\mathrm{Tm}}{l}}

C) \sqrt{\frac{\mathrm{m} l}{\mathrm{~T}}}

D) \sqrt{\frac{\mathrm{T}}{\mathrm{ml} l}}

258

Easy

A spherical body of mass 2 kg starting from rest acquires a kinetic energy of 10000 J at the end of $\mathrm{5^{th}}$ second. The force acted on the body is ________ N.

Options:

259

Medium

A particle performing uniform circular motion of radius $\frac{\pi}{2} \mathrm{~m} makes '\mathrm{x}' revolutions in time 't$'. Its tangential velocity is

Options:

A) \frac{\pi \mathrm{x}}{\mathrm{t}}

B) \frac{\pi x^2}{t}

C) \frac{\pi^2 x^2}{t}

D) \frac{\pi^2 \mathrm{x}}{\mathrm{t}}

260

Medium

A block of mass '$\mathrm{m}' (as shown in figure) moving with kinetic energy E compresses a spring through a distance 25 \mathrm{~cm} when, its speed is halved. The value of spring constant of used spring will be \mathrm{nE} \,\,\mathrm{Nm}^{-1} for \mathrm{n}=$ _________.

Options:

261

Medium

A body of mass 200 gram is tied to a spring of spring constant $12.5 \mathrm{~N} / \mathrm{m}, while other end of spring is fixed at point 'O'. If the body moves about 'O' in a circular path on a smooth horizontal surface with constant angular speed 5 \mathrm{~rad} / \mathrm{s}$ then the ratio of extension in the spring to its natural length will be

Options:

A) 1 : 2

B) 1 : 1

C) 2 : 3

D) 2 : 5

262

Easy

A uniform chain of 6 m length is placed on a table such that a part of its length is hanging over the edge of the table. The system is at rest. The co-efficient of static friction between the chain and the surface of the table is 0.5, the maximum length of the chain hanging from the table is ___________ m.

Options:

263

Medium

A particle of mass '$\mathrm{m}' moves along a circle of radius 'r' with constant tangential acceleration. If K.E. of the particle is 'E$' by the end of third revolution after beginning of the motion, then magnitude of tangential acceleration is

Options:

A) \frac{\mathrm{E}}{2 \pi \mathrm{rm}}

B) \frac{\mathrm{E}}{6 \pi \mathrm{rm}}

C) \frac{\mathrm{E}}{8 \pi \mathrm{rm}}

D) \frac{\mathrm{E}}{4 \pi \mathrm{rm}}

264

Easy

A 0.5 kg block moving at a speed of 12 ms$-1 compresses a spring through a distance 30 cm when its speed is halved. The spring constant of the spring will be _______________ Nm-$1.

Options:

265

Medium

A simple pendulum of length $2 \mathrm{~m} is given a horizontal push through angular displacement of 60^{\circ}. If the mass of bob is 200 gram, the angular velocity of the bob will be (Take Acceleration due to gravity =10 \mathrm{~m} / \mathrm{s}^2 ) \left(\sin 30^{\circ}=\cos 60^{\circ}=0.5, \cos 30^{\circ}=\sin 60^{\circ}=\sqrt{3} / 2\right)

Options:

A) 2 \sqrt{2} ~\mathrm{rad} / \mathrm{s}

B) 3 \sqrt{2} ~\mathrm{rad} / \mathrm{s}

C) 2 \sqrt{2.5} ~\mathrm{rad} / \mathrm{s}

D) 3 \sqrt{2.5} ~\mathrm{rad} / \mathrm{s}

266

Medium

A ball of mass 100 g is dropped from a height h = 10 cm on a platform fixed at the top of a vertical spring (as shown in figure). The ball stays on the platform and the platform is depressed by a distance ${h \over 2}. The spring constant is _____________ Nm-1. (Use g = 10 ms-$2)

Options:

267

Medium

A particle at rest starts moving with constant angular acceleration $4 ~\mathrm{rad} / \mathrm{s}^2$ in circular path. At what time the magnitudes of its tangential acceleration and centrifugal acceleration will be equal?

Options:

A) 0.4 s

B) 0.5 s

C) 0.8 s

D) 1.0 s

268

Medium

An engine is attached to a wagon through a shock absorber of length 1.5 m. The system with a total mass of 40,000 kg is moving with a speed of 72 kmh$-1 when the brakes are applied to bring it to rest. In the process of the system being brought to rest, the spring of the shock absorber gets compressed by 1.0 m. If 90% of energy of the wagon is lost due to friction, the spring constant is ____________ \times$ 105 N/m.

Options:

269

Medium

A bucket containing water is revolved in a vertical circle of radius $r. To prevent the water from falling down, the minimum frequency of revolution required is (\mathrm{g}=$ acceleration due to gravity)

Options:

A) 2 \pi \sqrt{\frac{\mathrm{r}}{\mathrm{g}}}

B) \frac{1}{2 \pi} \sqrt{\frac{\mathrm{r}}{\mathrm{g}}}

C) \frac{1}{2 \pi} \sqrt{\frac{\mathrm{g}}{\mathrm{r}}}

D) 2 \pi \sqrt{\frac{\mathrm{g}}{\mathrm{r}}}

270

Medium

A block moving horizontally on a smooth surface with a speed of 40 ms$-1 splits into two equal parts. If one of the parts moves at 60 ms-$1 in the same direction, then the fractional change in the kinetic energy will be x : 4 where x = ___________.

Options:

271

Medium

A body moving in a circular path with a constant speed has constant

Options:

A) momentum

B) velocity

C) acceleration

D) kinetic energy

272

Easy

Two persons A and B perform same amount of work in moving a body through a certain distance d with application of forces acting at angle 45$^\circ and 60^\circ with the direction of displacement respectively. The ratio of force applied by person A to the force applied by person B is {1 \over {\sqrt x }}$. The value of x is .................... .

Options:

273

Medium

Two bodies of masses '$\mathrm{m}' and '3 \mathrm{~m}' are rotating in horizontal speed of the body of mass 'm' is n times that of the value of heavier body; while the centripetal force is same for both. The value of n$ is

Options:

A) 3

B) 1

C) 9

D) 6

274

Medium

A uniform chain of length 3 meter and mass 3 kg overhangs a smooth table with 2 meter lying on the table. If k is the kinetic energy of the chain in joule as it completely slips off the table, then the value of k is ................. . (Take g = 10 m/s2)

Options:

275

Medium

A particle is moving along the circular path with constant speed and centripetal acceleration 'a'. If the speed is doubled, the ratio of its acceleration after and before the change is

Options:

A) 3: 1

B) 1: 4

C) 2: 1

D) 4: 1

276

Medium

A small block slides down from the top of hemisphere of radius R = 3 m as shown in the figure. The height 'h' at which the block will lose contact with the surface of the sphere is __________ m.(Assume there is no friction between the block and the hemisphere)

Options:

277

Medium

A body of mass 'm' is moving with speed 'V' along a circular path of radius 'r'. Now the speed is reduced to $\frac{V}{2}$ and radius is increased to '3r'. For this change, initial centripetal force needs to be

Options:

A) increased by $\frac{7}{12}$ times

B) increased by $\frac{10}{12}$ times

C) decreased by $\frac{11}{12}$ times

D) decreased by $\frac{1}{12}$ times

278

Easy

A force of F = (5y + 20)$\widehat j$ N acts on a particle. The work done by this force when the particle is moved from y = 0 m to y = 10 m is ___________ J.

Options:

279

Medium

A body attached to one end of a string performs motion along a vertical circle. Its centripetal acceleration, when the string is horizontal, will be [$\mathrm{g}=$ acceleration due to gravity]

Options:

A) zero

B) 5g

C) 3g

D) g

280

Medium

In a spring gun having spring constant 100 N/m a small ball 'B' of mass 100 g is put in its barrel (as shown in figure) by compressing the spring through 0.05 m. There should be a box placed at a distance 'd' on the ground so that the ball falls in it. If the ball leaves the gun horizontally at a height of 2 m above the ground. The value of d is _________ m. (g = 10 m/s2).

Options:

281

Medium

A projectile is thrown with an initial velocity $(a \hat{i}+b \hat{j}) \mathrm{m} / \mathrm{s}, where \hat{i} and \hat{j}$ are unit vectors along horizontal and vertical directions respectively. If the range of the projectile is twice the maximum height reached by it, then

Options:

A) \mathrm{b}=2 \mathrm{a}

B) \mathrm{b}=4 \mathrm{a}

C) \mathrm{b=\frac{a}{2}}

D) \mathrm{b}=\mathrm{a}

282

Easy

A ball of mass 4 kg, moving with a velocity of 10 ms$-1, collides with a spring of length 8 m and force constant 100 Nm-$1. The length of the compressed spring is x m. The value of x, to the nearest integer, is ____________.

Options:

283

Medium

A particle is performing U.C.M. along the circumference of a circle of diameter $50 \mathrm{~cm} with frequency 2 \mathrm{~Hz}. The acceleration of the particle in \mathrm{m} / \mathrm{s}^2$ is

Options:

A) 2 \pi^2

B) 4 \pi^2

C) 8 \pi^2

D) \pi^2

284

Easy

As shown in the figure, a particle of mass 10 kg is placed at a point A. When the particle is slightly displaced to its right, it starts moving and reaches the point B. The speed of the particle at B is x m/s. (Take g = 10 m/s2)The value of 'x' to the nearest integer is __________.

Options:

285

Medium

If $\omega_1 is angular velocity of hour hand of clock and \omega_2 is angular velocity of the earth, then the ratio \omega_1 : \omega_2$ is

Options:

A) 1 : 2

B) 2 : 3

C) 3 : 2

D) 2 : 1

286

Medium

The potential energy (U) of a diatomic molecule is a function dependent on r (interatomic distance) as $U = {\alpha \over {{r^{10}}}} - {\beta \over {{r^5}}} - 3where, \alpha and \beta are positive constants. The equilibrium distance between two atoms will be {\left( {{{2\alpha } \over \beta }} \right)^{{a \over b}}}$, where a = ___________.

Options:

287

Medium

The angular displacement of body performing circular motion is given by $\theta=5 \sin \frac{\pi t}{6}. The angular velocity of the body at t=3 second will be \left[\sin \frac{\pi}{2}=1, \cos \frac{\pi}{2}=0\right]

Options:

A) 5 \frac{\mathrm{rad}}{\mathrm{s}}

B) 1 \frac{\mathrm{rad}}{\mathrm{s}}

C) 2.5 \frac{\mathrm{rad}}{\mathrm{s}}

D) zero $\frac{\mathrm{rad}}{\mathrm{s}}

288

Medium

A body of mass 2 kg is driven by an engine delivering a constant power of 1 J/s. The body starts from rest and moves in a straight line. After 9 seconds, the body has moved a distance (in m) _______.

Options:

289

Medium

A body performing uniform circular motion of radius 'R' has frequency 'n'. It centripetal acceleration is

Options:

A) 8 $\pi^2nR^2

B) 4 $\pi^2n^2$R

C) 4 $\pi^2n^2R^2

D) 8 $\pi^2n^2$R

290

Medium

A block starts moving up an inclined plane of inclination 30o with an initial velocity of v0 . It comes back to its initial position with velocity ${{{v_0}} \over 2}. The value of the coefficient of kinetic friction between the block and the inclined plane is close to {I \over {1000}}$. The nearest integer to I is____.

Options:

291

Medium

The angle of banking '$\theta' for a meter gauge railway line is given by \theta=\tan ^{-1}\left(\frac{1}{20}\right)$. What is the elevation of the outer rail above the inner rail?

Options:

A) 20 \mathrm{~cm}

B) 10 \mathrm{~cm}

C) 0.2 \mathrm{~cm}

D) 5 \mathrm{~cm}

292

Medium

A cricket ball of mass 0.15 kg is thrown vertically up by a bowling machine so that it rises to a maximum height of 20 m after leaving the machine. If the part pushing the ball applies a constant force F on the ball and moves horizontally a distance of 0.2 m while launching the ball, the value of F (in N) is (g = 10 ms–2) ____.

Options:

293

Medium

A particle moves in a circular orbit of radius '$r' under a central attractive force, F=-\frac{k}{r}, where \mathrm{k}$ is a constant. The periodic time of its motion is proportional to

Options:

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

B) \mathrm{r}^{\frac{2}{3}}

C) r

D) r^{\frac{3}{2}}

294

Medium

A small block starts slipping down from a point B on an inclined plane AB, which is making an angle $\theta with the horizontal section BC is smooth and the remaining section CA is rough with a coefficient of friction \mu . It is found that the block comes to rest as it reaches the bottom (point A) of the inclined plane. If BC = 2AC, the coefficient of friction is given by \mu = ktan \theta $ . The value of k is _________.

Options:

295

Medium

A particle at rest starts moving with a constant angular acceleration of $4 \mathrm{~rad} / \mathrm{s}^2$ in a circular path. At what time the magnitude of its centripetal acceleration and tangential acceleration will be equal?

Options:

A) \frac{1}{4} \mathrm{~S}

B) \frac{2}{3} \mathrm{~S}

C) \frac{1}{2} \mathrm{~S}

D) \frac{1}{3} \mathrm{~S}

296

Medium

A particle (m = 1 kg) slides down a frictionless track (AOC) starting from rest at a point A (height 2 m). After reaching C, the particle continues to move freely in air as a projectile. When it reaching its highest point P (height 1 m), the kinetic energy of the particle (in )) is : (Figure drawn is schematic and not to scale; take g = 10 ms-2)

Options:

297

Medium

A child starts running from rest along a circular track of radius r with constant tangential acceleration a. After time t he feels that slipping of shoes on the ground has started. The coefficient of friction between shoes and the ground is [g = acceleration due to gravity]

Options:

A) \frac{\left[a^4 t^4+a^2 r^2\right]^{\frac{1}{2}}}{g r}

B) \frac{\left[a^4 t^4+a^2 r^2\right]}{r g}

C) \frac{\left[a^2 t^2+a^4 r^4\right]}{r g}

D) \frac{\left[a^4 t^4-a^2 r^2\right]^{\frac{1}{2}}}{r g}

298

Medium

A body is moving along a circular track of radius 100 m with velocity 20 \mathrm{~m} / \mathrm{s}. Its tangential acceleration is 3 \mathrm{~m} / \mathrm{s}^2, then its resultant acceleration will be

Options:

A) 5 \mathrm{~m} / \mathrm{s}^2

B) 4 \mathrm{~m} / \mathrm{s}^2

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

D) 3 \mathrm{~m} / \mathrm{s}^2

299

Medium

A particle starting from rest moves along the circumference of a circle of radius $r with angular acceleration \alpha. The magnitude of the average velocity, in the time it completes the small angular displacement \theta$ is

Options:

A) r\left(\frac{2}{\alpha \theta}\right)^2

B) r\left(\frac{\alpha \theta}{2}\right)^2

C) \rho\left(\frac{\alpha \theta}{2}\right)

D) r\left(\frac{\alpha \theta}{2}\right)^{\frac{1}{2}}

300

Medium

A particle of mass $m is performing UCM along a circle of radius r. The relation between centripetal acceleration a and kinetic energy E$ is given by

Options:

A) a=\frac{2 E}{m r}

B) a=2 E m

C) a=\frac{E}{m r}

D) a=\left(\frac{2 E}{m r}\right)^2

301

Medium

In non-uniform circular motion, the ratio of tangential to radial acceleration is ($r= radius, \alpha= angular acceleration and v=$ linear velocity)

Options:

A) \frac{r \alpha}{v}

B) \frac{v^2}{r a}

C) \frac{r \alpha^2}{v^2}

D) \frac{r^2 \alpha}{v^2}

302

Medium

A particle is moving in a radius $R with constant speed v$. The magnitude of average acceleration after half revolution is

Options:

A) \frac{2 \pi}{R v^2}

B) \frac{2 R}{\pi v}

C) \frac{2 v^2}{\pi R}

D) \frac{2 V}{\pi R^2}

303

Medium

A mass is whirled in a circular path with constant angular velocity and its linear velocity is v. If the string is now halved keeping the angular momentum same, the linear velocity is

Options:

A) 2 v

B) \frac{v}{2}

C) v

D) v \sqrt{2}

304

Medium

A body of mass m is performing a UCM in a circle of radius r with speed v. The work done by the centripetal force in moving it through \left(\frac{2}{3}\right) \mathrm{rd} of the circular path is

Options:

A) zero

B) m v^2 \pi r

C) \frac{2 \pi m v^2 r}{3}

D) \frac{2 m v^2 \pi}{3}

305

Medium

In U.C.M., when time interval \delta t \rightarrow 0, the angle between change in velocity ( \delta \mathbf{v} ) and linear velocity (\boldsymbol{v}) will be

Options:

A) 0^\circ

B) 90^\circ

C) 180^\circ

D) 45^\circ

306

Medium

A particle is performing U.C.M. along the circumference of a circle of diameter 50 cm with frequency 2 Hz . The acceleration of the particle in \mathrm{m} / \mathrm{s}^2 is

Options:

A) 2 \pi^2

B) 8 \pi^2

C) \pi^2

D) 4 \pi^2

307

Medium

A stone of mass 1 kg is tied to a string 2 m long and it's rotated at constant speed of 40 \mathrm{~ms}^{-1} in a vertical circle. The ratio of the tension at the top and the bottom is [Take g=10 \mathrm{~ms}^{-2}]

Options:

A) \frac{81}{79}

B) \frac{79}{81}

C) \frac{19}{12}

D) \frac{12}{19}

308

Medium

The real force ' F ' acting on a particle of mass m ' performing circular motion acts along the radius of circle ' r ' and is directed towards the centre of circle. The square root of magnitude of such force is ( T= periodic time)

Options:

A) \frac{2 \pi}{T} \sqrt{m r}

B) \frac{T m r}{4 \pi}

C) \frac{2 \pi T}{\sqrt{m r}}

D) \frac{T^2 m r}{4 \pi}

309

Medium

A cone filled with water is revolved in a vertical circle of radius $4 \mathrm{~m}$ and the water does not fall down. What must be the maximum period of revolution?

Options:

A) 2s

B) 4s

C) 1s

D) 6s

310

Medium

A train has to negotiate a curve of radius $400 \mathrm{~m}. By how much should the outer rail be raised with respect to the inner rail for a speed of 48 \mathrm{~km} / \mathrm{h} ? The distance between the rails is 1 \mathrm{~m}$.

Options:

A) 1.5 cm

B) 2.5 cm

C) 3.5 cm

D) 4.5 cm

310

Total Questions

Easy

234

Medium

Hard

Study Tips

Before You Start

• Review the chapter concepts thoroughly
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• Practice similar questions again later