Class 11 • Physics

Centre of Mass and Rotational Motion

Chapter-6

706 Questions

0 Video Solutions

88 Easy612 Medium6 Hard

Practice Questions

Click on any question to view the complete question with options and detailed solution.

Options:

A) Both Assertion and Reason are correct, Reason is the correct explanation of Assertion

B) Both Assertion and Reason are correct but Reason is not the correct explanation of Assertion

Options:

A) 1 : 1

B) 2 : 1

C) 1 : 4

D) 4 : 1

100

MediumJEE Mains2026

A uniform rod of mass m and length l suspended by means of two identical inextensible light strings as shown in figure. Tension in one string immediately after the other string is cut, is \_\_\_\_ . (g acceleration due to gravity)

Options:

A) \mathrm{mg} / \mathrm{s}

B) m g / 4

C) m g

D) m g / 2

101

EasyJEE Mains2022

Two billiard balls of mass 0.05 kg each moving in opposite directions with 10 ms$-$1 collide and rebound with the same speed. If the time duration of contact is t = 0.005 s, then what is the force exerted on the ball due to each other?

Options:

A) 100 N

B) 200 N

C) 300 N

D) 400 N

102

EasyJEE Mains2025

A rod of linear mass density 'λ' and length 'L' is bent to form a ring of radius 'R'. Moment of inertia of ring about any of its diameter is.

Options:

A) \frac{\lambda L^3}{8 \pi^2}

B) \frac{\lambda L^3}{16 \pi^2}

C) \frac{\lambda L^3}{4 \pi^2}

D) \frac{\lambda L^3}{12}

103

EasyJEE Mains2022

Two bodies A and B of masses 5 kg and 8 kg are moving such that the momentum of body B is twice that of the body A. The ratio of their kinetic energies will be :

Options:

A) 4 : 5

B) 2 : 5

C) 5 : 4

D) 5 : 2

104

EasyJEE Mains2025

Which of the following are correct expression for torque acting on a body? A. \vec{\tau}=\vec{r} \times \vec{L} B. \vec{\tau}=\frac{d}{d t}(\vec{r} \times \vec{p}) C. \vec{\tau}=\vec{r} \times \frac{d \vec{p}}{d t} D. \vec{\tau}=I \vec{\alpha} E. \vec{\tau}=\vec{r} \times \vec{F} ( \vec{r}= position vector; \vec{p}= linear momentum; \vec{L}= angular momentum; \vec{\alpha}= angular acceleration; I= moment of inertia; \vec{F}= force; t= time) Choose the correct answer from the options given below:

Options:

A) A, B, D and E Only

B) C and D Only

C) B, C, D and E Only

D) B, D and E Only

105

EasyJEE Mains2022

A body of mass M at rest explodes into three pieces, in the ratio of masses 1 : 1 : 2. Two smaller pieces fly off perpendicular to each other with velocities of 30 ms$-1 and 40 ms-$1 respectively. The velocity of the third piece will be :

Options:

A) 15 ms$-$1

B) 25 ms$-$1

C) 35 ms$-$1

D) 50 ms$-$1

106

EasyJEE Mains2025

If \vec{L} and \vec{P} represent the angular momentum and linear momentum respectively of a particle of mass ' m ' having position vector as \vec{r}=a(\hat{i} \cos \omega t+\hat{j} \sin \omega t). The direction of force is

Options:

A) Opposite to the direction of \vec{L}

B) Opposite to the direction of \vec{L} \times \vec{P}

C) Opposite to the direction of \vec{r}

D) Opposite to the direction of \vec{P}

107

EasyJEE Mains2022

Two blocks of masses 10 kg and 30 kg are placed on the same straight line with coordinates (0, 0) cm and (x, 0) cm respectively. The block of 10 kg is moved on the same line through a distance of 6 cm towards the other block. The distance through which the block of 30 kg must be moved to keep the position of centre of mass of the system unchanged is :

Options:

A) 4 cm towards the 10 kg block

B) 2 cm away from the 10 kg block

C) 2 cm towards the 10 kg block

D) 4 cm away from the 10 kg block

108

MediumJEE Mains2025

A force of 49 N acts tangentially at the highest point of a sphere (solid) of mass 20 kg , kept on a rough horizontal plane. If the sphere rolls without slipping, then the acceleration of the center of the sphere is

Options:

A) 2.5 \mathrm{~m} / \mathrm{s}^2

B) 3.5 \mathrm{~m} / \mathrm{s}^2

C) 0.25 \mathrm{~m} / \mathrm{s}^2

D) 0.35 \mathrm{~m} / \mathrm{s}^2

109

MediumJEE Mains2022

What percentage of kinetic energy of a moving particle is transferred to a stationary particle when it strikes the stationary particle of 5 times its mass? (Assume the collision to be head-on elastic collision)

Options:

A) 50.0%

B) 66.6%

C) 55.6%

D) 33.3%

110

MediumJEE Mains2025

The moment of inertia of a circular ring of mass M and diameter r about a tangential axis lying in the plane of the ring is :

Options:

A) \frac{3}{8} \mathrm{Mr}^2

B) 2 \mathrm{Mr}^2

C) \frac{1}{2} \mathrm{Mr}^2

D) \frac{3}{2} \mathrm{Mr}^2

111

EasyJEE Mains2022

An object is thrown vertically upwards. At its maximum height, which of the following quantity becomes zero?

Options:

A) Momentum

B) Potential Energy

C) Acceleration

D) Force

112

MediumJEE Mains2025

Moment of inertia of a rod of mass ' M ' and length ' L ' about an axis passing through its center and normal to its length is ' \alpha '. Now the rod is cut into two equal parts and these parts are joined symmetrically to form a cross shape. Moment of inertia of cross about an axis passing through its center and normal to plane containing cross is :

Options:

A) \alpha / 4

B) \alpha / 8

C) \alpha

D) \alpha / 2

113

MediumJEE Mains2021

A body of mass M moving at speed V0 collides elastically with a mass 'm' at rest. After the collision, the two masses move at angles $\theta1 and \theta2 with respect to the initial direction of motion of the body of mass M. The largest possible value of the ratio M/m, for which the angles \theta1 and \theta$2 will be equal, is :

Options:

A) 4

B) 1

C) 3

D) 2

114

EasyJEE Mains2025

A square Lamina OABC of length 10 cm is pivoted at ' \mathrm{O}^{\prime}. Forces act at Lamina as shown in figure. If Lamina remains stationary, then the magnitude of F is :

Options:

A) 10 N

B) 0 (zero)

C) 10\sqrt2 N

D) 20 N

115

MediumJEE Mains2021

Three objects A, B and C are kept in a straight line on a frictionless horizontal surface. The masses of A, B and C are m, 2m and 2m respectively. A moves towards B with a speed of 9 m/s and makes an elastic collision with it. Thereafter B makes a completely inelastic collision with C. All motions occur along same straight line. The final speed of C is :

Options:

A) 6 m/s

B) 9 m/s

C) 4 m/s

D) 3 m/s

116

MediumJEE Mains2025

A cord of negligible mass is wound around the rim of a wheel supported by spokes with negligible mass. The mass of wheel is 10 kg and radius is 10 cm and it can freely rotate without any friction. Initially the wheel is at rest. If a steady pull of 20 N is applied on the cord, the angular velocity of the wheel, after the cord is unwound by 1 m , would be:

Options:

A) 20 rad/s

B) 30 rad/s

C) 10 rad/s

D) 0 rad/s

117

EasyJEE Mains2021

Two billiard balls of equal mass 30g strike a rigid wall with same speed of 108 kmph (as shown) but at different angles. If the balls get reflected with the same speed then the ratio of the magnitude of impulses imparted to ball 'a' and ball 'b' by the wall along 'X' direction is :

Options:

A) 1 : 1

B) \sqrt 2 $ : 1

C) 2 : 1

D) 1 : $\sqrt 2

118

EasyJEE Mains2025

A uniform rod of mass 250 g having length 100 cm is balanced on a sharp edge at 40 cm mark. A mass of 400 g is suspended at 10 cm mark. To maintain the balance of the rod, the mass to be suspended at 90 cm mark, is

Options:

A) 290 g

B) 200 g

C) 190 g

D) 300 g

119

EasyJEE Mains2021

A bullet of '4 g' mass is fired from a gun of mass 4 kg. If the bullet moves with the muzzle speed of 50 ms$-$1, the impulse imparted to the gun and velocity of recoil of gun are :

Options:

A) 0.2 kg ms$-1, 0.1 ms-$1

B) 0.4 kg ms$-1, 0.05 ms-$1

C) 0.2 kg ms$-1, 0.05 ms-$1

D) 0.4 kg ms$-1, 0.1 ms-$1

120

MediumJEE Mains2025

A solid sphere and a hollow sphere of the same mass and of same radius are rolled on an inclined plane. Let the time taken to reach the bottom by the solid sphere and the hollow sphere be t_1 and t_2, respectively, then

Options:

A) t_1< t_2

B) t_1=2 t_2

C) t_1 >t_2

D) t_1=t_2

121

EasyJEE Mains2021

If the Kinetic energy of a moving body becomes four times its initial Kinetic energy, then the percentage change in its momentum will be :

Options:

A) 100%

B) 200%

C) 300%

D) 400%

122

EasyJEE Mains2025

A solid sphere is rolling without slipping on a horizontal plane. The ratio of the linear kinetic energy of the centre of mass of the sphere and rotational kinetic energy is :

Options:

A) \frac{3}{4}

B) \frac{4}{3}

C) \frac{5}{2}

D) \frac{2}{5}

123

EasyJEE Mains2021

An object of mass m1 collides with another object of mass m2, which is at rest. After the collision the objects move with equal speeds in opposite direction. The ratio of the masses m2 : m1 is :

Options:

A) 1 : 1

B) 3 : 1

C) 2 : 1

D) 1 : 2

124

MediumJEE Mains2025

A uniform solid cylinder of mass ' m ' and radius ' r ' rolls along an inclined rough plane of inclination 45^{\circ}. If it starts to roll from rest from the top of the plane then the linear acceleration of the cylinder's axis will be

Options:

A) \sqrt{2} \mathrm{~g}

B) \frac{1}{\sqrt{2}} \mathrm{~g}

C) \frac{1}{3 \sqrt{2}} \mathrm{~g}

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

125

MediumJEE Mains2021

Two identical blocks A and B each of mass m resting on the smooth horizontal floor are connected by a light spring of natural length L and spring constant K. A third block C of mass m moving with a speed v along the line joining A and B collides with A. The maximum compression in the spring is

Options:

A) \sqrt {{{mv} \over K}}

B) \sqrt {{m \over {2K}}}

C) \sqrt {{{mv} \over {2K}}}

D) v\sqrt {{m \over {2K}}}

126

MediumJEE Mains2025

A circular disk of radius R meter and mass M kg is rotating around the axis perpendicular to the disk. An external torque is applied to the disk such that \theta(t)=5 t^2-8 t, where \theta(t) is the angular position of the rotating disc as a function of time t. How much power is delivered by the applied torque, when t=2 \mathrm{~s} ?

Options:

A) 60 \mathrm{MR}^2

B) 72 \mathrm{MR}^2

C) 8 \mathrm{MR}^2

D) 108 \mathrm{MR}^2

127

MediumJEE Mains2021

A large block of wood of mass M = 5.99 kg is hanging from two long massless cords. A bullet of mass m = 10 g is fired into the block and gets embedded in it. The (block + bullet) then swing upwards, their centre of mass rising a vertical distance h = 9.8 cm before the (block + bullet) pendulum comes momentarily to rest at the end of its arc. The speed of the bullet just before collision is : (take g = 9.8 ms-2)

Options:

A) 831.4 m/s

B) 811.4 m/s

C) 841.4 m/s

D) 821.4 m/s

128

MediumJEE Mains2025

A solid sphere of mass ' m ' and radius ' r ' is allowed to roll without slipping from the highest point of an inclined plane of length ' L ' and makes an angle 30^{\circ} with the horizontal. The speed of the particle at the bottom of the plane is v_1. If the angle of inclination is increased to 45^{\circ} while keeping L constant. Then the new speed of the sphere at the bottom of the plane is v_2. The ratio v_1^2: v_2^2 is

Options:

A) 1: 3

B) 1: 2

C) 1: \sqrt{2}

D) 1: \sqrt{3}

129

MediumJEE Mains2021

Four equal masses, m each are placed at the comers of a square of length (l) as shown in the figure. The moment of inertia of the system about an axis passing through A and parallel to DB would be :

Options:

A) \sqrt 3 $ ml2

B) 2 ml2

C) ml2

D) 3 ml2

130

EasyJEE Mains2025

The torque due to the force (2 \hat{i}+\hat{j}+2 \hat{k}) about the origin, acting on a particle whose position vector is (\hat{i}+\hat{j}+\hat{k}), would be

Options:

A) \hat{j}+\hat{k}

B) \hat{i}-\hat{k}

C) \hat{i}-\hat{j}+\hat{k}

D) \hat{i}+\hat{k}

131

MediumJEE Mains2021

Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R.Assertion A : Body 'P' having mass M moving with speed 'u' has head-on collision elastically with another body 'Q' having mass 'm' initially at rest. If m << M, body 'Q' will have a maximum speed equal to '2u' after collision.Reason R : During elastic collision, the momentum and kinetic energy are both conserved.In the light of the above statements, choose the most appropriate answer from the options given below :

Options:

A) A is correct but R is not correct.

B) A is not correct but R is correct.

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

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

132

MediumJEE Mains2025

A uniform circular disc of radius ' \mathrm{R}^{\prime} and mass ' \mathrm{M}^{\prime} is rotating about an axis perpendicular to its plane and passing through its centre. A small circular part of radius R / 2 is removed from the original disc as shown in the figure. Find the moment of inertia of the remaining part of the original disc about the axis as given above.

Options:

A) \frac{17}{32} \mathrm{MR}^2

B) \frac{13}{32} \mathrm{MR}^2

C) \frac{9}{32} \mathrm{MR}^2

D) \frac{7}{32} \mathrm{MR}^2

133

MediumJEE Mains2021

A circular hole of radius $\left( {{a \over 2}} \right)$ is cut out of a circular disc of radius 'a' as shown in figure. The centroid of the remaining circular portion with respect to point 'O' will be :

Options:

A) {1 \over 6}a

B) {2 \over 3}a

C) {5 \over 6}a

D) {10 \over 11}a

134

MediumJEE Mains2024

A thin circular disc of mass $\mathrm{M} and radius \mathrm{R} is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with angular velocity \omega. If another disc of same dimensions but of mass \mathrm{M} / 2$ is placed gently on the first disc co-axially, then the new angular velocity of the system is :

Options:

A) \frac{4}{5} \omega

B) \frac{5}{4} \omega

C) \frac{3}{2} \omega

D) \frac{2}{3} \omega

135

MediumJEE Mains2020

Particle A of mass m1 moving with velocity $\left( {\sqrt3\widehat i + \widehat j} \right)m{s^{ - 1}} collides with another particle B of mass m2 which is at rest initially. Let \overrightarrow {{V_1}} and \overrightarrow {{V_2}} be the velocities of particles A and B after collision respectively. If m1 = 2m2 and after collision \overrightarrow {{V_1}} = \left( {\widehat i + \sqrt 3 \widehat j} \right) , the angle between \overrightarrow {{V_1}} and \overrightarrow {{V_2}} $ is :

Options:

A) 105o

B) 15o

C) -45o

D) 60o

136

MediumJEE Mains2024

Ratio of radius of gyration of a hollow sphere to that of a solid cylinder of equal mass, for moment of Inertia about their diameter axis $A B as shown in figure is \sqrt{8 / x}. The value of x$ is :

Options:

A) 34

B) 51

C) 67

D) 17

137

MediumJEE Mains2020

Blocks of masses m, 2m, 4m and 8m are arranged in a line on a frictionless floor. Another block of mass m, moving with speed v along the same line (see figure) collides with mass m in perfectly inelastic manner. All the subsequent collisions are also perfectly inelastic. By the time the last block of mass 8m starts moving the total energy loss is p% of the original energy. Value of 'p' is close to :

Options:

A) 37

B) 77

C) 87

D) 94

138

EasyJEE Mains2024

A disc of radius \mathrm{R} and mass \mathrm{M} is rolling horizontally without slipping with speed v. It then moves up an inclined smooth surface as shown in figure. The maximum height that the disc can go up the incline is :

Options:

A) \frac{3}{4} \frac{v^2}{\mathrm{~g}}

B) \frac{v^2}{g}

C) \frac{2}{3} \frac{v^2}{\mathrm{~g}}

D) \frac{1}{2} \frac{v^2}{\mathrm{~g}}

139

MediumJEE Mains2020

A block of mass 1.9 kg is at rest at the edge of a table, of height 1 m. A bullet of mass 0.1 kg collides with the block and sticks to it. If the velocity of the bullet is 20 m/s in the horizontal direction just before the collision then the kinetic energy just before the combined system strikes the floor, is [Take g = 10 m/s2 . Assume there is no rotational motion and loss of energy after the collision is negligable.]

Options:

A) 23 J

B) 21 J

C) 20 J

D) 19 J

140

MediumJEE Mains2024

A particle of mass $\mathrm{m} is projected with a velocity '\mathrm{u}' making an angle of 30^{\circ} with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at its maximum height \mathrm{h}$ is :

Options:

A) \frac{\mathrm{mu}^3}{\sqrt{2} \mathrm{~g}}

B) zero

C) \frac{\sqrt{3}}{2} \frac{\mathrm{mu}^2}{\mathrm{~g}}

D) \frac{\sqrt{3}}{16} \frac{\mathrm{mu}^3}{\mathrm{~g}}

141

MediumJEE Mains2020

A block of mass m = 1 kg slides with velocity v = 6 m/s on a frictionless horizontal surface and collides with a uniform vertical rod and sticks to it as shown. The rod is pivoted about O and swings as a result of the collision making angle $\theta before momentarily coming to rest. If the rod has mass M = 2 kg, and length l = 1 m, the value of \theta $ is approximately : (take g = 10 m/s2)

Options:

A) 63o

B) 69o

C) 55o

D) 49

142

EasyJEE Mains2024

A heavy iron bar of weight $12 \mathrm{~kg} is having its one end on the ground and the other on the shoulder of a man. The rod makes an angle 60^{\circ}$ with the horizontal, the weight experienced by the man is :

Options:

A) 3 \mathrm{~kg}

B) 6 \mathrm{~kg}

C) 6 \sqrt{3} \mathrm{~kg}

D) 12 \mathrm{~kg}

143

MediumJEE Mains2020

A particle of mass m with an initial velocity $u\widehat i collides perfectly elastically with a mass 3 m at rest. It moves with a velocity v\widehat j$ after collision, then, v is given by :

Options:

A) v = \sqrt {{2 \over 3}} u

B) v = {u \over {\sqrt 3 }}

C) v = {u \over {\sqrt 2 }}

D) v = {1 \over {\sqrt 6 }}u

144

MediumJEE Mains2023

A disc is rolling without slipping on a surface. The radius of the disc is $R. At t=0, the top most point on the disc is \mathrm{A}$ as shown in figure. When the disc completes half of its rotation, the displacement of point A from its initial position is

Options:

A) R\sqrt {({\pi ^2} + 1)}

B) 2R

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

D) 2R\sqrt {(1 + 4{\pi ^2})}

145

MediumJEE Mains2020

A rod of length L has non-uniform linear mass density given by $\rho (x) = a + b{\left( {{x \over L}} \right)^2} , where a and b are constants and 0 \le x \le $ L. The value of x for the centre of mass of the rod is at :

Options:

A) {3 \over 2}\left( {{{a + b} \over {2a + b}}} \right)L

B) {4 \over 3}\left( {{{a + b} \over {2a + 3b}}} \right)L

C) {3 \over 4}\left( {{{2a + b} \over {3a + b}}} \right)L

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

146

EasyJEE Mains2023

Given below are two statements: one is labelled as Assertion $\mathbf{A} and the other is labelled as Reason \mathbf{R}$ Assertion A : An electric fan continues to rotate for some time after the current is switched off. Reason R : Fan continues to rotate due to inertia of motion. In the light of above statements, choose the most appropriate answer from the options given below.

Options:

A) A is not correct but R is correct

B) A is correct but R is not correct

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

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

147

MediumJEE Mains2020

A particle of mass m is projected with a speed u from the ground at an angle $\theta = {\pi \over 3} w.r.t. horizontal (x-axis). When it has reached its maximum height, it collides completely inelastically with another particle of the same mass and velocity u\widehat i$ . The horizontal distance covered by the combined mass before reaching the ground is:

Options:

A) 2\sqrt 2 {{{u^2}} \over g}

B) {{3\sqrt 3 } \over 8}{{{u^2}} \over g}

C) {{3\sqrt 2 } \over 4}{{{u^2}} \over g}

D) {5 \over 8}{{{u^2}} \over g}

148

MediumJEE Mains2023

An object of mass 8 kg is hanging from one end of a uniform rod CD of mass 2 kg and length 1 m pivoted at its end C on a vertical wall as shown in figure. It is supported by a cable AB such that the system is in equilibrium. The tension in the cable is (Take g = 10 m/s$^2$)

Options:

A) 90 N

B) 240 N

C) 30 N

D) 300 N

149

MediumJEE Mains2020

Two particles of equal mass m have respective initial velocities $u\widehat i and u\left( {{{\widehat i + \widehat j} \over 2}} \right)$. They collide completely inelastically. The energy lost in the process is :

Options:

A) {1 \over 3}m{u^2}

B) {1 \over 8}m{u^2}

C) {3 \over 4}m{u^2}

D) \sqrt {{2 \over 3}} m{u^2}

150

EasyJEE Mains2022

The torque of a force $5 \hat{i}+3 \hat{j}-7 \hat{k} about the origin is \tau. If the force acts on a particle whose position vector is 2 i+2 j+k, then the value of \tau$ will be

Options:

A) 11 \hat{i}+19 \hat{j}-4 \hat{k}

B) -11 \hat{i}+9 \hat{j}-16 \hat{k}

C) -17 \hat{i}+19 \hat{j}-4 \hat{k}

D) 17 \hat{i}+9 \hat{j}+16 \hat{k}

151

MediumJEE Mains2020

A particle of mass m is dropped from a height h above the ground. At the same time another particle of the same mass is thrown vertically upwards from the ground with a speed of $\sqrt {2gh} . If they collide head-on completely inelastically, the time taken for the combined mass to reach the ground, in units of \sqrt {{h \over g}} $ is :

Options:

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

B) {1 \over 2}

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

D) \sqrt {{3 \over 4}}

152

MediumJEE Mains2022

A solid cylinder and a solid sphere, having same mass $M and radius R$, roll down the same inclined plane from top without slipping. They start from rest. The ratio of velocity of the solid cylinder to that of the solid sphere, with which they reach the ground, will be :

Options:

A) \sqrt{\frac{5}{3}}

B) \sqrt{\frac{4}{5}}

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

D) \sqrt{\frac{14}{15}}

153

MediumJEE Mains2020

As shown in figure, when a spherical cavity (centered at O) of radius 1 is cut out of a uniform sphere of radius R (centered at C), the centre of mass of remaining (shaded) part of sphere is at G, i.e, on the surface of the cavity. R can be detemined by the equation :

Options:

A) (R2 + R – 1) (2 – R) = 1

B) (R2 – R – 1) (2 – R) = 1

C) (R2 – R + 1) (2 – R) = 1

D) (R2 + R + 1) (2 – R) = 1

154

MediumJEE Mains2022

A spherical shell of 1 kg mass and radius R is rolling with angular speed $\omega on horizontal plane (as shown in figure). The magnitude of angular momentum of the shell about the origin O is {a \over 3} R2\omega$. The value of a will be :

Options:

A) 2

B) 3

C) 5

D) 4

155

MediumJEE Mains2020

The coordinates of centre of mass of a uniform flag shaped lamina (thin flat plate) of mass 4kg. (The coordinates of the same are shown in figure) are :

Options:

A) (0.75m, 1.75m)

B) (1m, 1.75m)

C) (0.75m, 0.75m)

D) (1.25m, 1.50m)

156

MediumJEE Mains2022

A ball is spun with angular acceleration $\alpha = 6t2 - 2t where t is in second and \alpha is in rads-2. At t = 0, the ball has angular velocity of 10 rads-$1 and angular position of 4 rad. The most appropriate expression for the angular position of the ball is :

Options:

A) {3 \over 2}{t^4} - {t^2} + 10t

B) {{{t^4}} \over 2} - {{{t^3}} \over 3} + 10t + 4

C) {{2{t^4}} \over 3} - {{{t^3}} \over 6} + 10t + 12

D) 2{t^4} - {{{t^3}} \over 2} + 5t + 4

157

MediumJEE Mains2020

Three point particles of masses 1.0 kg, 1.5 kg and 2.5 kg are placed at three corners of a right angle triangle of sides 4.0 cm, 3.0 cm and 5.0 cm as shown in the figure. The center of mass of the system is at a point:

Options:

A) 2.0 cm right and 0.9 cm above 1 kg mass

B) 0.9 cm right and 2.0 cm above 1 kg mass

C) 0.6 cm right and 2.0 cm above 1 kg mass

D) 1.5 cm right and 1.2 cm above 1 kg mass

158

MediumJEE Mains2022

A $\sqrt {34} m long ladder weighing 10 kg leans on a frictionless wall. Its feet rest on the floor 3 m away from the wall as shown in the figure. If Ef and Fw are the reaction forces of the floor and the wall, then ratio of {F_w}/{F_f}$ will be : (Use g = 10 m/s2.)

Options:

A) {6 \over {\sqrt {110} }}

B) {3 \over {\sqrt {113} }}

C) {3 \over {\sqrt {109} }}

D) {2 \over {\sqrt {109} }}

159

MediumJEE Mains2019

Three particles of masses 50 g, 100 g and 150 g are placed at the vertices of an equilateral triangle of side 1 m (as shown in the figure). The (x, y) coordinates of the centre of mass will be :

Options:

A) \left( {{{\sqrt 3 } \over 4}m,{5 \over {12}}m} \right)

B) \left( {{7 \over {12}}m,{{\sqrt 3 } \over 8}m} \right)

C) \left( {{7 \over {12}}m,{{\sqrt 3 } \over 4}m} \right)

D) \left( {{{\sqrt 3 } \over 8}m,{7 \over {12}}m} \right)

160

EasyJEE Mains2022

Match List-I with List-II List-I List-II (A) Moment of inertia of solid sphere of radius R about any tangent. (I) ${5 \over 3}M{R^2} (B) Moment of inertia of hollow sphere of radius (R) about any tangent. (II) {7 \over 5}M{R^2} (C) Moment of inertia of circular ring of radius (R) about its diameter. (III) {1 \over 4}M{R^2} (D) Moment of inertia of circular disc of radius (R) about any diameter. (IV) {1 \over 2}M{R^2}$ Choose the correct answer from the options given below :

Options:

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

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

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

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

161

MediumJEE Mains2019

A man (mass = 50 kg) and his son (mass = 20 kg) are standing on a frictionless surface facing each other. The man pushes his son so that he starts moving at a speed of 0.70 ms–1 with respect to the man. The speed of the man with respect to the surface is :

Options:

A) 0.28 ms–1

B) 0.47 ms–1

C) 0.20 ms–1

D) 0.14 ms–1

162

MediumJEE Mains2022

One end of a massless spring of spring constant k and natural length l0 is fixed while the other end is connected to a small object of mass m lying on a frictionless table. The spring remains horizontal on the table. If the object is made to rotate at an angular velocity $\omega$ about an axis passing through fixed end, then the elongation of the spring will be :

Options:

A) {{k - m{\omega ^2}{l_0}} \over {m{\omega ^2}}}

B) {{m{\omega ^2}{l_0}} \over {k + m{\omega ^2}}}

C) {{m{\omega ^2}{l_0}} \over {k - m{\omega ^2}}}

D) {{k + m{\omega ^2}{l_0}} \over {m{\omega ^2}}}

163

MediumJEE Mains2019

Two particles, of masses M and 2M, moving, as shown, with speeds of 10 m/s and 5 m/s, collide elastically at the origin. After the collision, they move along the indicated directions with speeds u1 and u2, respectively. The values of u1 and u2 are nearly :

Options:

A) 6.5 m/s and 3.2 m/s

B) 6.5 m/s and 6.3 m/s

C) 3.2 m/s and 6.3 m/s

D) 3.2 m/s and 12.6 m/s

164

EasyJEE Mains2022

A solid spherical ball is rolling on a frictionless horizontal plane surface about its axis of symmetry. The ratio of rotational kinetic energy of the ball to its total kinetic energy is

Options:

A) {2 \over 5}

B) {2 \over 7}

C) {1 \over 5}

D) {7 \over 10}

165

MediumJEE Mains2019

A particle of mass 'm' is moving with speed '2v' and collides with a mass '2m' moving with speed 'v' in the same direction. After collision, the first mass is stopped completely while the second one splits into two particles each of mass 'm', which move at angle 45° with respect to the origianl direction. The speed of each of the moving particle will be :-

Options:

A) 2 $\sqrt2$v

B) v / (2 $\sqrt2$ )

C) v / $\sqrt2

D) \sqrt2$v

166

MediumJEE Mains2022

A thin circular ring of mass M and radius R is rotating with a constant angular velocity 2 rads$-1 in a horizontal plane about an axis vertical to its plane and passing through the center of the ring. If two objects each of mass m be attached gently to the opposite ends of a diameter of ring, the ring will then rotate with an angular velocity (in rads-$1).

Options:

A) {M \over {(M + m)}}

B) {{(M + 2m)} \over {2M}}

C) {{2M} \over {(M + 2m)}}

D) {{2(M + 2m)} \over M}

167

MediumJEE Mains2019

A wedge of mass M = 4m lies on a frictionless plane. A particle of mass m approaches the wedge with speed v. There is no friction between the particle and the plane or between the particle and the wedge. The maximum height climbed by the particle on the wedge is given by :-

Options:

A) {{{v^2}} \over {g}}

B) {{2{v^2}} \over {7g}}

C) {{{v^2}} \over {2g}}

D) {{2{v^2}} \over {5g}}

168

EasyJEE Mains2022

If force $\overrightarrow F = 3\widehat i + 4\widehat j - 2\widehat k acts on a particle position vector 2\widehat i + \widehat j + 2\widehat k$ then, the torque about the origin will be :

Options:

A) 3\widehat i + 4\widehat j - 2\widehat k

B) - 10\widehat i + 10\widehat j + 5\widehat k

C) 10\widehat i + 5\widehat j - 10\widehat k

Options:

A) {g \over {13l}}

B) {g \over {2l}}

C) {g \over {3l}}

D) {7g \over {3l}}

301

MediumMHT CET2025

Three rods of same mass are placed as shown in figure. The co-ordinates of centre of mass of the system are

Options:

A) \left(\frac{\mathrm{a}}{3}, \frac{\mathrm{a}}{3}\right)

B) \left(a, \frac{a}{2}\right)

C) \left(2 a \frac{a}{2}\right)

D) \left(\frac{2 a}{3}, \frac{a}{3}\right)

302

MediumJEE Mains2019

Two identical spherical balls of mass M and radius R each are stuck on two ends of a rod of length 2R and mass M (see figure). The moment of inertia of the system about the axis passing perpendicularly through the centre of the rod is :

Options:

A) {{17} \over {15}}$ MR2

B) {{137} \over {15}}$ MR2

C) {{209} \over {15}}$ MR2

D) {{152} \over {15}}$ MR2

303

MediumMHT CET2025

A sphere of mass ' m ', moving with velocity ' 3 u ' collides head-on with another identical sphere at rest. If ' e ' is coefficient of restitution then what will be the ratio of velocity of the second sphere to that of first sphere after collision?

Options:

A) \frac{1-\mathrm{e}}{1+\mathrm{e}}

B) \frac{1+\mathrm{e}}{1-\mathrm{e}}

C) \frac{e+1}{e-1}

D) \frac{e-1}{e+1}

304

MediumJEE Mains2019

To mop-clean a floor, a cleaning machine presses a circular mop of radius R vertically down with a total force F and rotates it with a constant angular speed about its axis. If the force F is distributed uniformly over the mop and if coefficient of friction between the mop and the floor is $\mu $, the torque, applied by the machine on the mop is -

Options:

A) \mu $FR/2

B) \mu $FR/3

C) \mu $FR/6

D) {2 \over 3}\mu $FR

305

MediumMHT CET2025

An object of mass ' m ' moving with velocity ' u ' collides with another stationary object of mass ' M ' and stops just after the collision. The coefficient of restitution is

Options:

A) \frac{\mathrm{m}}{\mathrm{M}+\mathrm{m}}

B) \frac{M-m}{M+m}

C) \frac{\mathrm{m}}{\mathrm{M}}

D) 1

306

MediumJEE Mains2019

A homogeneous solid cylindrical roller of radius R and mass M is pulled on a cricket pitch by a horizontal force. Assuming rolling without slipping, angular acceleration of the cylinder is -

Options:

A) {F \over {2mR}}

B) {2F \over {3mR}}

C) {3F \over {2mR}}

D) {F \over {3mR}}

307

MediumMHT CET2025

There is head-on elastic collision between the two particles moving in the same direction with speeds 5 \mathrm{~m} / \mathrm{s} and 3 \mathrm{~m} / \mathrm{s} respectively. After collision, the velocity of the first particle becomes 4 \mathrm{~m} / \mathrm{s} in the same direction. The velocity of the second particle should be

Options:

A) 6 \mathrm{~m} / \mathrm{s} in the same direction.

B) 4 \mathrm{~m} / \mathrm{s} in the same direction.

C) 2 \mathrm{~m} / \mathrm{s} in the opposite direction.

D) 3 \mathrm{~m} / \mathrm{s} in the same direction.

308

MediumJEE Mains2019

A rod of length 50 cm is pivoted at one end. It is raised such that if makes an angle of 30o from the horizontal as shown and released from rest. Its angular speed when it passes through the horizontal (in rad s$-1) will be (g = 10 ms-$2)

Options:

A) \sqrt {{{30} \over 7}}

B) \sqrt {30}

C) {{\sqrt {20} } \over 3}

D) {{\sqrt {30} } \over 2}

309

MediumMHT CET2024

Three identical metal balls each of radius ' r ' are placed such that an equilateral triangle is formed when centres of three ball are joined. The centre of mass of the system is located at

Options:

A) centre of one of the balls.

B) point of intersection of medians.

C) line joining centres of any two balls.

D) on the circumference of any one of the balls.

310

MediumJEE Mains2019

An L-shaped object, made of thin rods of uniform mass density, is suspended with a string as shown in figure. If AB = BC, and the angle made by AB with downward vertical is $\theta $, thrown :

Options:

A) tan$\theta = {1 \over {2\sqrt 3 }}

B) tan$\theta = {1 \over 2}

C) tan$\theta = {2 \over {\sqrt 3 }}

D) tan$\theta = {1 \over 3}

311

MediumMHT CET2024

In case of system of two-particles of different masses, the centre of mass lies

Options:

A) at the mid-point of line joining the two particles.

B) on the line joining the two particles.

C) at one end of the line joining the two particles.

D) on the line perpendicular to the line joining the two particles.

312

MediumJEE Mains2019

If the angular momentum of a planet of mass m, moving around the Sun in a circular orbit is L, about the center of the Sun, its areal velocity is :

Options:

A) {L \over m}

B) {4L \over m}

C) {L \over 2m}

D) {2L \over m}

313

MediumMHT CET2024

A particle of mass m collides with another stationary particle of mass M. The particle m stops just after collision. The coefficient of restitution is

Options:

A) \frac{m}{M}

B) \frac{M-m}{M+m}

C) 1

D) \frac{\mathrm{m}}{\mathrm{M}+\mathrm{m}}

314

MediumJEE Mains2018

A thin circular disk is in the xy plane as shown in the figure. The ratio of its moment of inertia about z and z' axes will be :

Options:

A) 1 : 3

B) 1 : 4

C) 1 : 5

D) 1 : 2

315

MediumMHT CET2024

1000 small balls, each weighing 1 gram, strike one square cm of area per second with a velocity 50 \mathrm{~m} / \mathrm{s} in a normal direction and rebound with the same velocity. The value of pressure on the surface will be

Options:

A) 10^7 \mathrm{~N} / \mathrm{m}^2

B) 10^6 \mathrm{~N} / \mathrm{m}^2

C) 5 \times 10^6 \mathrm{~N} / \mathrm{m}^2

D) 2 \times 10^6 \mathrm{~N} / \mathrm{m}^2

316

MediumJEE Mains2018

From a uniform circular disc of radius R and mass 9M, a small disc of radius R/3 is removed as shown in the figure. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through centre of disc is :

Options:

A) {{37} \over 9}M{R^2}

B) 4M{R^2}

C) {{40} \over 9}M{R^2}

D) 10M{R^2}

317

MediumMHT CET2024

A moving body with mass ' \mathrm{m}_1 ' strikes a stationary mass ' \mathrm{m}_2 '. What should be the ratio \frac{m_1}{m_2} so as to decrease the velocity of first by (1.5) times the velocity after the collision?

Options:

A) 1: 25

B) 1: 5

C) 5: 1

D) 25: 1

318

MediumJEE Mains2018

Seven identical circular planar disks, each of mass M and radius R are welded symmetrically as shown. The moment of inertia of the arrangement about the axis normal to the plane and passing through the point P is :

Options:

A) {{181} \over 2}M{R^2}

B) {{55} \over 2}M{R^2}

C) {{19} \over 2}M{R^2}

D) {{73} \over 2}M{R^2}

319

MediumMHT CET2024

A metal rod of weight ' W ' is supported by two parallel knife-edges A and B . The rod is in equilibrium in horizontal position. The distance ' between two knife-edges is ' r '. The centre of mass of the rod is at a distance ' x ' from A. The normal reaction on A is

Options:

A) \frac{\mathrm{W} \cdot \mathrm{r}}{\mathrm{x}}

B) \frac{\mathrm{W} \cdot \mathrm{x}}{\mathrm{r}}

C) \mathrm{\frac{W \cdot(r-x)}{x}}

D) \mathrm{\frac{W \cdot(r-x)}{r}}

320

MediumJEE Mains2018

A thin uniform bar of length $L and mass 8 m lies on a smooth horizontal table. Two point masses m and 2 m are moving in the same horizontal plane from opposite sides of the bar with speeds 2\upsilon and \upsilon respectively. The masses stick to the bar after collision at a distance {L \over 3} and {L \over 6}$ respectively from the center of the bar. If the br starts rotating about its center of mass as a result of collision, the angular speed of the bar will be :

Options:

A) {v \over {5L}}

B) {6v \over {5L}}

C) {3v \over {5L}}

D) {v \over {6L}}

321

MediumMHT CET2024

In the system of two particles of masses ' \mathrm{m}_1 ' and ' \mathrm{m}_2 ', the first particle is moved by a distance 'd' towards the centre of mass. To keep the centre of mass unchanged, the second particle will have to be moved by a distance

Options:

A) \frac{\mathrm{m}_2}{\mathrm{~m}_1} \mathrm{~d}, towards the centre of mass.

B) \frac{\mathrm{m}_1}{\mathrm{~m}_2} \mathrm{~d}, away from the centre of mass.

C) \frac{\mathrm{m}_1}{\mathrm{~m}_2} \mathrm{~d}, towards the centre of mass.

D) \frac{\mathrm{m}_2}{\mathrm{~m}_1} \mathrm{~d}, away from the centre of mass.

322

MediumJEE Mains2018

A thin rod MN, free to rotate in the vertical plane aboutthe fixed end N, is held horizontal. When the end M is released the speed of this end, when the rod makes an angle $\alpha $ with the horizontal, will be proportional to : (see figure)

Options:

A) \sqrt {\sin \alpha }

B) {\sin \alpha }

C) \sqrt {\cos \alpha }

D) {\cos \alpha }

323

MediumMHT CET2024

In projectile motion two particles of masses \mathrm{m}_1 and m_2 have velocities \vec{V}_1, and \vec{V}_2 respectively at time t=0. Their velocities become \overline{V_1^{\prime}} and \overrightarrow{V_2^{\prime}} at time 2 t while still moving in air. The value of \left[\left(m_1 \overrightarrow{V_1^{\prime}}+m_2 \overrightarrow{V_2^{\prime}}\right)-\left(m_1 \vec{V}_1+m_2 \vec{V}_2\right)\right] is ( \mathrm{g}= acceleration due to gravity)

Options:

A) zero

B) \frac{1}{2}\left(\mathrm{~m}_1+\mathrm{m}_2\right) \mathrm{gt}

C) \left(m_1+m_2\right) g t

D) 2\left(m_1+m_2\right) g t

324

MediumJEE Mains2018

A uniform rod $AB is suspended from a point X, at a variable distance x from A, as shown, To make the rod horizontal, a mass m is suspended from its end A.A set of (m,x)$ values is recorded. The appropriate variables that give a straight line, when plotted, are :

Options:

A) m,x

B) m,{1 \over x}

C) m,{1 \over {{x^2}}}

D) m,{x^2}

325

MediumMHT CET2024

A meter scale is supported on a wedge at its centre of gravity. A body of weight ' w ' is suspended from the 20 cm mark and another weight of 25 gram is suspended from 74 cm mark balances it and the meter scale remains perfectly horizontal. Neglecting the weight of the meter scale, the weight of the body is

Options:

A) 33 gram wt.

B) 30 gram wt.

C) 20 gram wt.

D) 15 gram wt.

326

MediumJEE Mains2018

A force of $40 N acts on a point B at the end of an L-shaped object, as shown in the figure. The angle \theta that will produce maximum moment of the force about point A$ is given by :

Options:

A) \tan \theta = {1 \over 2}

B) \tan \theta = 2

C) \tan \theta = 4

D) \tan \theta = {1 \over 4}

327

MediumMHT CET2023

A person with machine gun can fire 50 g bullets with a velocity of $240 \mathrm{~m} / \mathrm{s}. A 60 \mathrm{~kg} tiger moves towards him with a velocity of 12 \mathrm{~m} / \mathrm{s}$. In order to stop the tiger in track, the number of bullets the person fires towards the tiger is

Options:

A) 50

B) 60

C) 70

D) 80

328

MediumJEE Mains2017

The machine as shown has 2 rods of length1 m connected by a pivot at the top. The end of one rod is connected to the floor by a stationary pivot and the end of the other rod has a roller that rolls along the floor in a slot. As the roller goes back and forth, a 2 kg weight moves up and down. If the roller is moving towards right at a constant speed, the weight moves up with a :

Options:

A) Constant speed

B) decreasing speed

C) increasing speed

D) speed which is ${3 \over 4}$th of that of the roller when the weight is 0.4 m above the ground

329

MediumMHT CET2023

A simple spring has length $l and force constant K. It is cut in to two springs of length l_1 and l_2 such that l_1=n l_2(n is an integer). The force constant of spring of length l_1$ is

Options:

A) K(1+n)

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

C) K

D) \frac{K}{(n+1)}

330

MediumJEE Mains2017

A circular hole of radius ${R \over 4}$ is made in a thin uniform disc having mass M and radius R, as shown in figure. The moment of inertia of the remaining portion of the disc about an axis passing through the point O and perpendicular to the plane of the disc is :

Options:

A) {{219\,M{R^2}} \over {256}}

B) {{237\,M{R^2}} \over {512}}

C) {{19\,M{R^2}} \over {512}}

D) {{197\,M{R^2}} \over {256}}

331

MediumMHT CET2023

A particle of mass '$m' moving east ward with a speed 'v' collides with another particle of same mass moving north-ward with same speed 'v'. The two particles coalesce after collision. The new particle of mass '2 \mathrm{~m}' will move in north east direction with a speed (in \mathrm{m} / \mathrm{s}$ )

Options:

A) \mathrm{V}

B) 2 \mathrm{~V}

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

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

332

MediumJEE Mains2017

A uniform disc of radius R and mass M is free to rotate only about its axis. A string is wrapped over its rim and a body of mass m is tied to the free end of the string as shown in the figure. The body is released from rest. Then the acceleration of the body is :

Options:

A) {{2\,\,mg} \over {2\,m + M}}

B) {{2\,\,Mg} \over {2\,m + M}}

C) {{2\,\,mg} \over {2\,M + m}}

D) {{2\,\,Mg} \over {2\,M + M}}

333

MediumMHT CET2023

A ball kept at $20 \mathrm{~m} height falls freely in vertically downward direction and hits the ground. The coefficient of restitution is 0.4. Velocity of the ball first rebound is \left[\mathrm{g}=10 \mathrm{~ms}^{-2}\right]

Options:

A) 4 \mathrm{~ms}^{-1}

B) 8 \mathrm{~ms}^{-1}

C) 12 \mathrm{~ms}^{-1}

D) 16 \mathrm{~ms}^{-1}

334

MediumJEE Mains2017

Moment of inertia of an equilateral triangular lamina ABC, about the axis passing through its centre O and perpendicular to its plane is Io as shown in the figure. A cavity DEF is cut out from the lamina, where D, E, F are the mid points of the sides. Moment of inertia of the remaining part of lamina about the same axis is :

Options:

A) {7 \over 8}$ Io

B) {15 \over 16}$ Io

C) {{3\,{{\rm I}_o}} \over 4}

D) {{31\,{{\rm I}_o}} \over 32}

335

MediumMHT CET2023

A mass '$\mathrm{M}' moving with velocity '\mathrm{V}' along \mathrm{X}-axis collides and sticks to another mass 2 \mathrm{M} which is moving along \mathrm{Y}-axis with velocity '3 \mathrm{~V}$'. The velocity of the combination after collision is

Options:

A) \frac{V}{3} \hat{i}+2 V \hat{j}

B) \frac{V}{2} \hat{i}+V \hat{j}

C) \frac{V}{3} \hat{i}-2 V \hat{j}

D) \frac{V}{2} \hat{i}-V \hat{j}

336

MediumJEE Mains2017

In a physical balance working on the principle of moments, when 5 mg weight is placed on the left pan, the beam becomes horizontal. Both the empty pans of the balance are of equal mass. Which of the following statements is correct ?

Options:

A) Left arm is longer than the right arm

B) Both the arms are of same length

C) Left arm is shorter than the right arm

D) Every object that is weighed using this balance appears lighter than its actual weight.

337

MediumMHT CET2023

Consider the following statements $\mathrm{A} and \mathrm{B}$. Identify the correct choice in the given answers. A. In an inelastic collision, there is no loss in kinetic energy during collision. B. During a collision, the linear momentum of the entire system of particles is conserved if there is no external force acting on the system.

Options:

A) Both $\mathrm{A} and \mathrm{B}$ are wrong.

B) Both A and B are correct.

C) \mathrm{A} is wrong and \mathrm{B}$ is correct.

D) A is correct and B is wrong.

338

MediumJEE Mains2017

A slender uniform rod of mass M and length $l is pivoted at one end so that it can rotate in a vertical plane (see figure). There is negligible friction at the pivot. The free end is held vertically above the pivot and then released. The angular acceleration of the rod when it makes an angle \theta$ with the vertical is

Options:

A) {{2g} \over {3l}}\cos \theta

B) {{3g} \over {2l}}\sin \theta

C) {{2g} \over {3l}}\sin \theta

D) {{3g} \over {3l}}\sin \theta

339

MediumMHT CET2023

A body falls on a surface of coefficient of restitution 0.6 from a height of $1 \mathrm{~m}$. Then the body rebounds to a height of

Options:

A) 1 m

B) 0.36 m

C) 0.4 m

D) 0.6 m

340

MediumJEE Mains2017

The moment of inertia of a uniform cylinder of length $l and radius R about its perpendicular bisector is I. What is the ratio {l \over R}$ such that the moment of inertia is minimum?

Options:

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

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

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

D) 1

341

MediumMHT CET2022

Two massless springs of spring constant $\mathrm{K}_1 and \mathrm{K}_2 are connected one after the other forming a single chain, suspended vertically and certain mass is attached to the free end. If 'e_1' and 'e_2' are their respective extensions and '\mathrm{f}$' is their stretching force, the total extension produced is

Options:

A) \mathrm{f}\left(\frac{1}{\mathrm{~K}_1}+\frac{1}{\mathrm{~K}_2}\right)

B) \mathrm{f}\left(\frac{1}{\mathrm{~K}_1}-\frac{1}{\mathrm{~K}_2}\right)

C) \mathrm{f}\left(\mathrm{K}_1+\mathrm{K}_2\right)

D) \mathrm{f}\left(\mathrm{K}_1-\mathrm{K}_2\right)

342

MediumJEE Mains2016

Concrete mixture is made by mixing cement, stone and sand in a rotating cylindrical drum. If the drum rotates too fast, the ingredients remain stuck to the wall of the drum and proper mixing of ingredients does not take place. The maximum rotational speed of the drum in revolutions per minute(rpm) to ensure proper mixing is close to : (Take the radius of the drum to be 1.25 m and its axle to be horizontal) :

Options:

A) 0.4

B) 1.3

C) 8.0

D) 27.0

343

MediumMHT CET2021

A wooden black of mass '$\mathrm{m}' moves with velocity '\mathrm{V}' and collides with another block of mass '4 \mathrm{~m}', which is at rest. After collision the block of mass '\mathrm{m}$' comes to rest. The coefficient of restitution will be

Options:

A) 0.7

B) 0.25

C) 0.4

D) 0.5

344

MediumJEE Mains2016

A cubical block of side 30 cm is moving with velocity 2 ms−1 on a smooth horizontal surface. The surface has a bump at a point O as shown in figure. The angular velocity (in rad/s) of the block immediately after it hits the bump, is :

Options:

A) 5.0

B) 6.7

C) 9.4

D) 13.3

345

MediumMHT CET2021

Force is applied to a body of mass $2 \mathrm{~kg} at rest on a frictionless horizontal surface as shown in the force against time (F-t)$ graph. The speed of the body after 1 second is

Options:

A) 7.5 m/s

B) 12.5 m/s

C) 10 m/s

D) 15 m/s

346

MediumJEE Mains2016

A roller is made by joining together two cones at their vertices $0. It is kept on two rails AB and CD, which are placed asymmetrically (see figure), with its axis perpendicular to CD and its center O at the center of line joining AB and CD (see figure). It is given a light push so that it starts rolling with its center O moving parallel to CD$ in the direction shown. As it moves, the roller will tend to :

Options:

A) go straight

B) turn left and right alternately

C) turn left

D) turn right

347

MediumMHT CET2021

A molecule of mass 'm' moving with velocity 'v' makes 5 elastic collisions with a wall of container per second. The change in momentum of the wall per second in 5 collisions will be

Options:

A) 10 mv

B) 5 mv

C) \frac{1}{5}$ mv

D) \frac{1}{10}$ mv

348

MediumJEE Mains2015

From a solid sphere of mass $M and radius R$ a cube of maximum possible volume is cut. Moment of inertia of cube about an axis passing through its center and perpendicular to one of its face is:

Options:

A) {{4M{R^2}} \over {9\sqrt {3\pi } }}

B) {{4M{R^2}} \over {3\sqrt {3\pi } }}

C) {{M{R^2}} \over {32\sqrt {2\pi } }}

D) {{M{R^2}} \over {16\sqrt {2\pi } }}

349

MediumMHT CET2021

A particle of mass '$m' collides with another stationary particle of mass 'M'. A particle of mass '\mathrm{m}$' stops just after collision. The coefficient of restitution is

Options:

A) \frac{M}{m}

B) \frac{m+M}{M}

C) \frac{M-m}{M+m}

D) \frac{\mathrm{m}}{\mathrm{M}}

350

MediumJEE Mains2014

A bob of mass $m attached to an inextensible string of length l is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed \omega \,rad/s$ about the vertical. About the point of suspension:

Options:

A) angular momentum is conserved

B) angular momentum changes in magnitude but not in direction.

C) angular momentum changes in direction but not in magnitude.

D) angular momentum changes both in direction and magnitude.

351

MediumMHT CET2021

Two masses '$m_{\mathrm{a}}' and '\mathrm{m}_{\mathrm{b}}' moving with velocities 'v_{\mathrm{a}}' and 'v_{\mathrm{b}}' opposite directions collide elastically. Alter the collision 'm_a' and 'm_b' move with velocities and 'v_{\mathrm{b}}' and 'v_a' respectively, then the ratio \mathrm{m_a:m_b}$ is

Options:

A) \frac{v_a+v_b}{v_a-v_b}

B) \frac{1}{2}

C) 1

D) \frac{v_a-v_b}{v_a+v_b}

352

MediumJEE Mains2014

A mass $'m' is supported by a massless string wound around a uniform hollow cylinder of mass m and radius R.$ If the string does not slip on the cylinder, with what acceleration will the mass fall or release?

Options:

A) {{2g} \over 3}

B) {{g} \over 2}

C) {{5g} \over 6}

D) g

353

MediumMHT CET2020

In system of two particles of masses m_1 and m_2, the first particle is moved by a distance d towards the centre of mass. To keep the centre of mass unchanged, the second particle will have to be moved by a distance

Options:

A) \frac{m_2}{m_1} d, towards the centre of mass

B) \frac{m_1}{m_2} d, away from the centre of mass

C) \frac{m_1}{m_2} d, towards the centre of mass

D) \frac{m_2}{m_1} d, away from the centre of mass

354

MediumJEE Mains2013

A hoop of radius $r and mass m rotating with an angular velocity {\omega _0}$ is placed on a rough horizontal surface. The initial velocity of the center of the hoop is zero. What will be the velocity of the center of the hoop when it cases to slip?

Options:

A) {{r{\omega _0}} \over 4}

B) {{r{\omega _0}} \over 3}

C) {{r{\omega _0}} \over 2}

D) {r{\omega _0}}

355

MediumMHT CET2020

N number of balls of mass m \mathrm{~kg} moving along positive direction of X - axis, strike a wall per second and return elastically. The velocity of each ball is u \mathrm{~m} / \mathrm{s}. The force exerted on the wall by the balls in newton, is

Options:

A) 0

B) 2 m N u

C) \frac{m N u}{2}

D) m N u

356

MediumJEE Mains2011

A pulley of radius $2 m is rotated about its axis by a force F = \left( {20t - 5{t^2}} \right) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10kg-{m^2}$ the number of rotation made by the pulley before its direction of motion is reversed, is:

Options:

A) more than $3 but less than 6

B) more than $6 but less than 9

C) more than $9

D) less than $3

357

MediumMHT CET2020

A batsman hits a ball of mass 0.2 kg straight towards the bowler without changing its initial speed of $6 \mathrm{~m} / \mathrm{s}$. What is the impulse imparted to the ball?

Options:

A) 3.2 N-s

B) 1.6 N-s

C) 4 N-s

D) 2.4 N-s

358

MediumJEE Mains2011

A mass $m hangs with the help of a string wrapped around a pulley on a frictionless bearing. The pulley has mass m and radius R. Assuming pulley to be a perfect uniform circular disc, the acceleration of the mass m,$ if the string does not slip on the pulley, is:

Options:

A) g

B) {2 \over 3}g

C) {g \over 3}

D) {3 \over 2}g

359

MediumMHT CET2020

A bullet of mass $m moving with velocity v is fired into a wooden block of mass M$, If the bullet remains embedded in the block, the final velocity of the system is

Options:

A) \frac{m+M}{m}

B) \frac{M+m}{m v}

C) \frac{m v}{m+M}

D) \frac{v}{m(M+m)}

360

MediumJEE Mains2011

A thin horizontal circular disc is rotating about a vertical axis passing through its center. An insect is at rest at a point near the rim of the disc. The insect now moves along a diameter of the disc to reach its other end. During the journey of the insect, the angular speed of the disc.

Options:

A) continuously decreases

B) continuously increases

C) first increases and then decreases

D) remains unchanged

361

MediumMHT CET2019

A block of mass ' m ' moving on a frictionless surface at speed ' v ' collides elastically with a block of same mass, initially at rest. Now the first block moves at an angle ' \theta ' with its initial direction and has speed ' v_1 '. The speed of the second block after collision is

Options:

A) \sqrt{v_1^2-v^2}

B) \sqrt{v^2-v_1^2}

C) \sqrt{v^2+v_1^2}

D) \sqrt{v-v_1}

362

MediumJEE Mains2010

A small particle of mass $m is projected at an angle \theta with the x-axis with an initial velocity {v_0} in the x-y plane as shown in the figure. At a time t < {{{v_0}\sin \theta } \over g}, the angular momentum of the particle is ................, where \widehat i,\widehat j and \widehat k are unit vectors along x,y and z$-axis respectively.

Options:

A) - mg\,{v_0}{t^2}\cos \theta \widehat j

B) mg\,{v_0}t\cos \theta \widehat k

C) - {1 \over 2}mg\,{v_0}{t^2}\cos \,\theta \widehat k

D) {1 \over 2}mg\,{v_0}{t^2}\cos \theta \widehat i

363

MediumVITEEE2023

A particle of mass $10 \mathrm{~kg} moving eastwards with a speed 10 \mathrm{~m} / \mathrm{s} collides with another particle of the same mass moving northward with the same speed 10 \mathrm{~m} / \mathrm{s}. The two particles coalesce on collision. The new particle of mass 20 \mathrm{~kg}$ will move in the north-east direction with velocity.

Options:

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

B) 5 \mathrm{~m} / \mathrm{s}

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

D) None

364

MediumJEE Mains2009

A thin uniform rod of length $l and mass m is swinging freely about a horizontal axis passing through its end. Its maximum angular speed is \omega $. Its center of mass rises to a maximum height of:

Options:

A) {1 \over 6}\,\,{{l\omega } \over g}

B) {1 \over 2}\,\,{{{l^2}{\omega ^2}} \over g}

C) {1 \over 6}\,\,{{{l^2}{\omega ^2}} \over g}

D) {1 \over 3}\,\,{{{l^2}{\omega ^2}} \over g}

365

MediumVITEEE2022

A projectile is launched vertically from the earth with speed $v_1 hits a satellite at the height h moving with speed v_2. If both have the same mass m$, then what is the common velocity if they move together after the collision?

Options:

A) v_1+v_2

B) \sqrt{v_1^2+v_2^2}

C) \frac{\sqrt{v_1^2+v_2^2}}{2}

D) 2 \sqrt{v_1^2+v_2^2}

366

MediumJEE Mains2008

Consider a uniform square plate of side $' a ' and mass 'm'$. The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is

Options:

A) {5 \over 6}m{a^2}

B) {1 \over 12}m{a^2}

C) {7 \over 12}m{a^2}

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

367

MediumJEE Mains2007

For the given uniform square lamina $ABCD, whose center is O,

Options:

A) {I_{AC}} = \sqrt 2 \,\,{I_{EF}}

B) \sqrt 2 {I_{AC}} = {I_{EF}}

C) {I_{AD}} = 3{I_{EF}}

D) {I_{AC}} = {I_{EF}}

368

MediumJEE Mains2007

Angular momentum of the particle rotating with a central force is constant due to

Options:

A) constant torque

B) constant force

C) constant linear momentum

D) zero torque

369

MediumJEE Mains2007

A round uniform body of radius $R, mass M and moment of inertia I rolls down (without slipping) an inclined plane making an angle \theta $ with the horizontal. Then its acceleration is

Options:

A) {{g\,\sin \theta } \over {1 - M{R^2}/I}}

B) {{g\,\sin \theta } \over {1 + I/M{R^2}}}

C) {{g\,\sin \theta } \over {1 + M{R^2}/I}}

D) {{g\,\sin \theta } \over {1 - I/M{R^2}}}

370

MediumJEE Mains2006

A force of $ - F\widehat k acts on O, the origin of the coordinate system. The torque about the point (1, -1)$ is

Options:

A) F\left( {\widehat i - \widehat j} \right)

B) - F\left( {\widehat i + \widehat j} \right)

C) F\left( {\widehat i + \widehat j} \right)

D) - F\left( {\widehat i - \widehat j} \right)

371

MediumJEE Mains2006

Four point masses, each of value $m, are placed at the corners of a square ABCD of side l. The moment of inertia of this system about an axis passing through A and parallel to BD$ is

Options:

A) 2m{l^2}

B) \sqrt 3 m{l^2}

C) 3m{l^2}

D) m{l^2}

372

MediumJEE Mains2006

A thin circular ring of mass $m and radius R is rotating about its axis with a constant angular velocity \omega . Two objects each of mass M are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with an angular velocity \omega ' =

Options:

A) {{\omega \left( {m + 2M} \right)} \over m}

B) {{\omega \left( {m - 2M} \right)} \over {\left( {m + 2M} \right)}}

C) {{\omega m} \over {\left( {m + M} \right)}}

D) {{\omega m} \over {\left( {m + 2M} \right)}}

373

MediumJEE Mains2005

An annular ring with inner and outer radii ${R_1} and {R_2} is rolling without slipping with a uniform angular speed. The ratio of the forces experienced by the two particles situated on the inner and outer parts of the ring, {{{F_1}} \over {{F_2}}}\,$ is

Options:

A) {\left( {{{{R_1}} \over {{R_2}}}} \right)^2}

B) {{{{R_2}} \over {{R_1}}}}

C) {{{{R_1}} \over {{R_2}}}}

D) 1

374

MediumJEE Mains2005

The moment of inertia of a uniform semicircular disc of mass $M and radius r$ about a line perpendicular to the plane of the disc through the center is

Options:

A) {2 \over 5}M{r^2}

B) {1 \over 4}Mr

C) {1 \over 2}M{r^2}

D) M{r^2}

375

MediumJEE Mains2004

One solid sphere $A and another hollow sphere B are of same mass and same outer radii. Their moment of inertia about their diameters are respectively {I_A} and {I_B}$ such that

Options:

A) {I_A} < {I_B}

B) {I_A} > {I_B}

C) {I_A} = {I_B}

D) {{{I_A}} \over {{I_B}}} = {{{d_A}} \over {{d_B}}} where {d_A} and {d_B}$ are their densities.

376

MediumJEE Mains2004

A solid sphere is rotating in free space. If the radius of the sphere is increased keeping mass same which on of the following will not be affected ?

Options:

A) Angular velocity

B) Angular momentum

C) Moment of inertia

D) Rotational kinetic energy

377

MediumJEE Mains2003

A circular disc $X of radius R is made from an iron plate of thickness t, and another disc Y of radius 4 R is made from an iron plate of thickness {t \over 4}. Then the relation between the moment of inertia {I_X} and {I_Y}$ is

Options:

A) {I_Y} = 32{I_X}

B) {I_Y} = 16{I_X}

C) {I_Y} = {I_X}

D) {I_Y} = 64{I_X}

378

MediumJEE Mains2003

A particle performing uniform circular motion has angular frequency is doubled & its kinetic energy halved, then the new angular momentum is

Options:

A) {L \over 4}

B) 2L

C) 4L

D) {L \over 2}

379

MediumJEE Mains2003

Let $\overrightarrow F be the force acting on a particle having position vector \overrightarrow r , and \overrightarrow \tau $ be the torque of this force about the origin. Then

Options:

A) \overrightarrow {r.} \overrightarrow \tau = 0\,\, and \overrightarrow {F.} \overrightarrow \tau \ne 0\,\,

B) \overrightarrow {r.} \vec \tau \ne 0{\mkern 1mu} {\mkern 1mu} and \overrightarrow {F.} \overrightarrow \tau = 0\,\,

C) \overrightarrow {r.} \vec \tau \ne 0{\mkern 1mu} and \overrightarrow {F.} \overrightarrow \tau \ne 0

D) \overrightarrow {r.} \vec \tau = 0{\mkern 1mu} and \overrightarrow {F.} \overrightarrow \tau = 0\,\,

380

MediumJEE Mains2002

Moment of inertia of a circular wire of mass $M and radius R$ about its diameter is

Options:

A) {{M{R^2}} \over 2}

B) M{R^2}

C) 2M{R^2}

D) {{M{R^2}} \over 4}

381

MediumJEE Mains2002

A solid sphere, a hollow sphere and a ring are released from top of an inclined plane (frictionless) so that they slide down the plane. Then maximum acceleration down the plane is for (no rolling)

Options:

A) solid sphere

B) hollow sphere

C) ring

D) all same

382

MediumJEE Mains2002

A particle of mass $m moves along line PC with velocity v$ as shown. What is the angular momentum of the particle about P?

Options:

A) mvL

B) mvl

C) mvr

D) zero

383

MediumJEE Mains2002

Initial angular velocity of a circular disc of mass $M is {\omega _1}. Then two small spheres of mass m$ are attached gently to diametrically opposite points on the edge of the disc. What is the final angular velocity of the disc?

Options:

A) \left( {{{M + m} \over M}} \right)\,\,{\omega _1}

B) \left( {{{M + m} \over m}} \right)\,\,{\omega _1}

C) \left( {{M \over {M + 4m}}} \right)\,\,{\omega _1}

Options:

A) m g r

B) m g y

C) m g \sqrt{r^2-y^2}

D) m g r \sqrt{1-\left(\frac{r}{y}\right)^2}

706

Total Questions

Easy

612

Medium

Hard

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