Question 1: Palette 1/169

JEE Mains 2026 MCQ

When the position vector $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ changes sign as $-\vec{r}$, which one of the following vector will not flip under sign change?

A

Velocity

B

Linear momentum

C

Acceleration

D

Angular momentum

Question 2: Palette 2/169

JEE Mains 2026 MCQ

Two circular discs of radius each 10 cm are joined at their centres by a rod of length 30 cm and mass 600 gm as shown in figure.

If the mass of each disc is 600 gm and applied torque between two discs is $43 \times 10^5$ dyne.cm, the angular acceleration of the discs about the given axis $A B$ is $\_\_\_\_$ $\mathrm{rad} / \mathrm{s}^2$.

A

100

B

22

C

27

D

11

Question 3: Palette 3/169

JEE Mains 2026 MCQ

A thin uniform rod $(X)$ of mass $M$ and length $L$ is pivoted at a height $\left(\frac{L}{3}\right)$ as shown in the figure. The rod is allowed to fall from a vertical position and lie horizontally on the table. The angular velocity of this rod when it hits the table top, is $\_\_\_\_$ .

( $\mathrm{g}=$ gravitational acceleration)

A

$\sqrt{\frac{3}{2} \frac{g}{L}}$

B

$\sqrt{\frac{3 g}{L}}$

C

$\frac{3}{\sqrt{2}} \sqrt{\frac{g}{L}}$

D

$\frac{1}{\sqrt{2}} \sqrt{\frac{g}{L}}$

Question 4: Palette 4/169

JEE Mains 2026 MCQ

Two masses 400 g and 350 g are suspended from the ends of a light string passing over a heavy pulley of radius 2 cm . When released from rest the heavier mass is observed to fall 81 cm in 9 s . The rotational inertia of the pulley is $\_\_\_\_$ $\mathrm{kg} \cdot \mathrm{m}^2$. $\left(\mathrm{g}=9.8 \mathrm{~m} / \mathrm{s}^2\right)$

A

$8.3 \times 10^{-3}$

B

$4.75 \times 10^{-3}$

C

$1.86 \times 10^{-2}$

D

$9.5 \times 10^{-3}$

Question 5: Palette 5/169

JEE Mains 2026 MCQ

Two small balls with masses $m$ and 2 m are attached to both ends of a rigid rod of length $d$ and negligible mass. If angular momentum of this system is $L$ about an axis (A) passing through its centre of mass and perpendicular to the rod then angular velocity of the system about $A$ is :

A

$\frac{4}{3} \frac{L}{m d^2}$

B

$\frac{3}{2} \frac{L}{m d^2}$

C

$\frac{2 L}{5 m d^2}$

D

$\frac{2 L}{m d^2}$

Question 6: Palette 6/169

JEE Mains 2026 MCQ

The moment of inertia of a square loop made of four uniform solid cylinders, each having radius $R$ and length $L(\mathrm{R}<\mathrm{L})$ about an axis passing through the mid points of opposite sides, is (Take the mass of the entire loop as $M$ ) :

A

$\frac{3}{4} M R^2+\frac{1}{6} M L^2$

B

$\frac{3}{8} M R^2+\frac{7}{12} M L^2$

C

$\frac{3}{4} M R^2+\frac{7}{12} M L^2$

D

$\frac{3}{8} M R^2+\frac{1}{6} M L^2$

Question 7: Palette 7/169

JEE Mains 2026 MCQ

A uniform bar of length 12 cm and mass 20 m lies on a smooth horizontal table. Two point masses $m$ and $2 m$ are moving in opposite directions with same speed of $v$ and in the same plane as the bar, as shown in figure. These masses strike the bar simultaneously and get stuck to it. After collision the entire system is rotating with angular frequency $\omega$. The ratio of $v$ and $\omega$ is :

A

33

B

$2 \sqrt{88}$

C

32

D

66

Question 8: Palette 8/169

JEE Mains 2026 MCQ

A cylindrical tube $A B$ of length $l$, closed at both ends contains an ideal gas of 1 mol having molecular weight $M$. The tube is rotated in a horizontal plane with constant angular velocity $\omega$ about an axis perpendicular to $A B$ and passing through the edge at end $A$, as shown in the figure. If $P_A$ and $P_B$ are the pressures at $A$ and $B$ respectively, then (Consider the temperature is same at all points in the tube)

A

${P}_{{B}}={P}_{{A}} {e}^{\left(\frac{M \omega^2 {l}^2}{3 R T}\right)}$

B

$P_B=P_A$

C

${P}_{{B}}={P}_{{A}} {e}^{\left(\frac{{M} \omega^2 {l}^2}{2 {R T}}\right)}$

D

$P_B=P_A e^{\left(\frac{M \omega^2 l^2}{R T}\right)}$

Question 9: Palette 9/169

JEE Mains 2026 MCQ

A solid sphere of mass 5 kg and radius 10 cm is kept in contact with another solid sphere of mass 10 kg and radius 20 cm . The moment of inertia of this pair of spheres about the tangent passing through the point of contact is $\_\_\_\_$ $\mathrm{kg} \cdot \mathrm{m}^2$.

A

0.18

B

0.72

C

0.36

D

0.63

Question 10: Palette 10/169

JEE Mains 2026 MCQ

The pulley shown in figure is made using a thin rim and two rods of length equal to diameter of the rim. The rim and each rod have a mass of M. Two blocks of mass of M and m are attached to two ends of a light string passing over the pulley, which is hinged to rotate freely in vertical plane about its center. The magnitudes of the acceleration experienced by the blocks is ________ (assume no slipping of string on pulley).

A

$ \dfrac{(M-m) g}{\left[\left(\frac{8}{3}\right) M+m\right]} $

B

$ \dfrac{(M - m)g}{2M + m} $

C

$ \dfrac{(M - m)g}{M + m} $

D

$ \dfrac{(M-m) g}{\left[\left(\frac{13}{6}\right) M+m\right]} $

Question 11: Palette 11/169

JEE Mains 2026 MCQ

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)

A

$\mathrm{mg} / \mathrm{s}$

B

$m g / 4$

C

$m g$

D

$m g / 2$

Question 12: Palette 12/169

JEE Mains 2026 MCQ

A solid cylinder having radius $R$ and length $L$ is slipping on a rough horizontal plane. At time $t=0$ the cylinder has a translational velocity $v_{\mathrm{o}}=49 \mathrm{~m} / \mathrm{s}$, perpendicular to its axis and a rotational velocity $v_{\mathrm{o}} / 4 R$ about the centre. The time taken by the cylinder to start rolling is $\_\_\_\_$ seconds. (coefficient of kinetic friction $\mu_K=0.25$ and $g=9.8 \mathrm{~m} / \mathrm{s}^2$ )

A

15

B

5

C

10

D

7.5

Question 13: Palette 13/169

JEE Mains 2026 MCQ

A solid sphere $(A)$ of mass $5 m$ and a spherical shell $(B)$ of mass $m$, both having same radius, are placed on a rough surface. When a force of same magnitude is applied tangentially at the highest points of $A$ and $B$, they start rolling without slipping with an acceleration of $a_A$ and $a_B$, respectively. The ratio of $a_A$ and $a_B$ is $\_\_\_\_$ .

A

$ 5: 21 $

B

$ 6: 10 $

C

$ 21: 25 $

D

$ 1: 5 $

Question 14: Palette 14/169

JEE Mains 2026 MCQ

A solid sphere of radius 4 cm and mass 5 kg is rotating (rotation axis is passing through the centre of the sphere) with an angular velocity of 1200 rpm . It is brought to rest in 10 s by applying a constant torque. The torque applied and the number of rotations it made before it comes to rest are $\_\_\_\_$ and $\_\_\_\_$ respectively.

A

$0.128 \pi \mathrm{Nm}, 100$

B

$0.0128 \pi \mathrm{Nm}, 50$

C

$0.128 \pi \mathrm{Nm}, 50$

D

$0.0128 \pi \mathrm{Nm}, 100$

Question 15: Palette 15/169

JEE Mains 2026 MCQ

A wheel initially at rest is subjected to a uniform angular acceleration about its axis. In the first 2 s it rotates through an angle $\theta_1$ and in the next 2 s it rotates through an angle $\theta_2$. The ratio $\frac{\theta_2}{\theta_1}$ is $\_\_\_\_$ .

A

6

B

3

C

4

D

${\frac{1}{3}}$

Question 16: Palette 16/169

JEE Mains 2026 MCQ

An object of uniform density rolls up the curved path with the initial velocity $v_{\mathrm{o}}$ as shown in the figure. If the maximum height attained by an object is $\frac{7 v_0^2}{10 \mathrm{~g}}$ ( $\mathrm{g}=$ acceleration due to gravity), the object is a $\_\_\_\_$ .

A

solid cylinder

B

ring

C

disc

D

solid sphere

Question 17: Palette 17/169

JEE Mains 2026 MCQ

A solid sphere of mass $M$ and radius $R$ is divided into two unequal parts. The smaller part having mass $M / 8$ is converted into a sphere of radius $r$ and the larger part is converted into a circular disc of thickness $t$ and radius $2 R$. If $I_1$ is moment of inertia of a sphere having radius $r$ about an axis through its centre and $I_2$ is the moment of inertia of a disc about its diameter, the ratio of their moment of inertia $I_2 / I_1=$ $\_\_\_\_$

A

35

B

70

C

140

D

210

Question 18: Palette 18/169

JEE Mains 2026 MCQ

The position of an object having mass 0.1 kg as a function of time $t$ is given as

$\vec{r} = \left( 10 t^2 \hat{i} + 5 t^3 \hat{j} \right)$ m. At $t = 1$ s, which of the following statements are correct?

A. The linear momentum $\vec{p} = \left( 2 \hat{i} + 1.5 \hat{j} \right)$ kg·m/s.

B. The force acting on the object $\vec{F} = \left( 2 \hat{i} + 3 \hat{j} \right)$ N.

C. The angular momentum of the object about its origin $\vec{L} = 15 \hat{k}$ J·s.

D. The torque acting on the object about its origin $\vec{\tau} = 20 \hat{k}$ N·m.

Choose the correct answer from the options given below:

A

A, B and C only

B

B, C and D only

C

A, C and D only

D

A, B and D only

Question 19: Palette 19/169

JEE Mains 2025 MCQ

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.

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} $

Question 20: Palette 20/169

JEE Mains 2025 MCQ

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:

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

Question 21: Palette 21/169

JEE Mains 2025 MCQ

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

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 $\vec{P}$

Question 22: Palette 22/169

JEE Mains 2025 MCQ

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

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 $

Question 23: Palette 23/169

JEE Mains 2025 MCQ

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 :

A

$\frac{3}{8} \mathrm{Mr}^2$

B

$2 \mathrm{Mr}^2$

C

$\frac{1}{2} \mathrm{Mr}^2$

D

$\frac{3}{2} \mathrm{Mr}^2$

Question 24: Palette 24/169

JEE Mains 2025 MCQ

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 :

A

$\alpha / 4$

B

$\alpha / 8$

C

$\alpha$

D

$\alpha / 2$

Question 25: Palette 25/169

JEE Mains 2025 MCQ

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 :

A

10 N

B

0 (zero)

C

10$\sqrt2$ N

D

20 N

Question 26: Palette 26/169

JEE Mains 2025 MCQ

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:

A

20 rad/s

B

30 rad/s

C

10 rad/s

D

0 rad/s

Question 27: Palette 27/169

JEE Mains 2025 MCQ

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

A

290 g

B

200 g

C

190 g

D

300 g

Question 28: Palette 28/169

JEE Mains 2025 MCQ

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

A

$t_1< t_2$

B

$t_1=2 t_2$

C

$t_1 >t_2$

D

$t_1=t_2$

Question 29: Palette 29/169

JEE Mains 2025 MCQ

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 :

A

$\frac{3}{4}$

B

$\frac{4}{3}$

C

$\frac{5}{2}$

D

$\frac{2}{5}$

Question 30: Palette 30/169

JEE Mains 2025 MCQ

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

A

$\sqrt{2} \mathrm{~g}$

B

$\frac{1}{\sqrt{2}} \mathrm{~g}$

C

$\frac{1}{3 \sqrt{2}} \mathrm{~g}$

D

$\frac{\sqrt{2} g}{3}$

Question 31: Palette 31/169

JEE Mains 2025 MCQ

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 $t=2 \mathrm{~s}$ ?

A

$60 \mathrm{MR}^2$

B

$72 \mathrm{MR}^2$

C

$8 \mathrm{MR}^2$

D

$108 \mathrm{MR}^2$

Question 32: Palette 32/169

JEE Mains 2025 MCQ

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

A

$1: 3$

B

$1: 2$

C

$1: \sqrt{2}$

D

$1: \sqrt{3}$

Question 33: Palette 33/169

JEE Mains 2025 MCQ

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

A

$\hat{j}+\hat{k}$

B

$\hat{i}-\hat{k}$

C

$\hat{i}-\hat{j}+\hat{k}$

D

$\hat{i}+\hat{k}$

Question 34: Palette 34/169

JEE Mains 2025 MCQ

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.

A

$\frac{17}{32} \mathrm{MR}^2$

B

$\frac{13}{32} \mathrm{MR}^2$

C

$\frac{9}{32} \mathrm{MR}^2$

D

$\frac{7}{32} \mathrm{MR}^2$

Question 35: Palette 35/169

JEE Mains 2024 MCQ

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 :

A

$\frac{4}{5} \omega$

B

$\frac{5}{4} \omega$

C

$\frac{3}{2} \omega$

D

$\frac{2}{3} \omega$

Question 36: Palette 36/169

JEE Mains 2024 MCQ

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 :

A

34

B

51

C

67

D

17

Question 37: Palette 37/169

JEE Mains 2024 MCQ

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 :

A

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

B

$\frac{v^2}{g}$

C

$\frac{2}{3} \frac{v^2}{\mathrm{~g}}$

D

$\frac{1}{2} \frac{v^2}{\mathrm{~g}}$

Question 38: Palette 38/169

JEE Mains 2024 MCQ

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 :

A

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

B

zero

C

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

D

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

Question 39: Palette 39/169

JEE Mains 2024 MCQ

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 :

A

$3 \mathrm{~kg}$

B

$6 \mathrm{~kg}$

C

$6 \sqrt{3} \mathrm{~kg}$

D

$12 \mathrm{~kg}$

Question 40: Palette 40/169

JEE Mains 2023 MCQ

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

A

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

B

$2R$

C

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

D

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

Question 41: Palette 41/169

JEE Mains 2023 MCQ

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.

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

Question 42: Palette 42/169

JEE Mains 2023 MCQ

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$^2$)

A

90 N

B

240 N

C

30 N

D

300 N

Question 43: Palette 43/169

JEE Mains 2022 MCQ

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

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

$17 \hat{i}+9 \hat{j}+16 \hat{k}$

Question 44: Palette 44/169

JEE Mains 2022 MCQ

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 :

A

$\sqrt{\frac{5}{3}}$

B

$\sqrt{\frac{4}{5}}$

C

$\sqrt{\frac{3}{5}}$

D

$\sqrt{\frac{14}{15}}$

Question 45: Palette 45/169

JEE Mains 2022 MCQ

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 :

A

2

B

3

C

5

D

4

Question 46: Palette 46/169

JEE Mains 2022 MCQ

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 :

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

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

Question 47: Palette 47/169

JEE Mains 2022 MCQ

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 E_f and F_w are the reaction forces of the floor and the wall, then ratio of ${F_w}/{F_f}$ will be :

(Use g = 10 m/s².)

A

${6 \over {\sqrt {110} }}$

B

${3 \over {\sqrt {113} }}$

C

${3 \over {\sqrt {109} }}$

D

${2 \over {\sqrt {109} }}$

Question 48: Palette 48/169

JEE Mains 2022 MCQ

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 :

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

Question 49: Palette 49/169

JEE Mains 2022 MCQ

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 :

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

${{k + m{\omega ^2}{l_0}} \over {m{\omega ^2}}}$

Question 50: Palette 50/169

JEE Mains 2022 MCQ

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

A

${2 \over 5}$

B

${2 \over 7}$

C

${1 \over 5}$

D

${7 \over 10}$

Question 51: Palette 51/169

JEE Mains 2022 MCQ

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).

A

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

B

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

C

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

D

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

Question 52: Palette 52/169

JEE Mains 2022 MCQ

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 :

A

$3\widehat i + 4\widehat j - 2\widehat k$

B

$ - 10\widehat i + 10\widehat j + 5\widehat k$

C

$10\widehat i + 5\widehat j - 10\widehat k$

D

$10\widehat i + \widehat j - 5\widehat k$

Question 53: Palette 53/169

JEE Mains 2021 MCQ

A system consists of two identical spheres each of mass 1.5 kg and radius 50 cm at the end of light rod. The distance between the centres of the two spheres is 5 m. What will be the moment of inertia of the system about an axis perpendicular to the rod passing through its midpoint?

A

18.75 kgm²

B

1.905 $\times$ 10⁵ kgm²

C

19.05 kgm²

D

1.875 $\times$ 10⁵ kgm²

Question 54: Palette 54/169

JEE Mains 2021 MCQ

Angular momentum of a single particle moving with constant speed along circular path :

A

changes in magnitude but remains same in the direction

B

remains same in magnitude and direction

C

remains same in magnitude but changes in the direction

D

is zero

Question 55: Palette 55/169

JEE Mains 2021 MCQ

Two discs have moments of inertia I₁ and I₂ about their respective axes perpendicular to the plane and passing through the centre. They are rotating with angular speeds, $\omega$₁ and $\omega$₂ respectively and are brought into contact face to face with their axes of rotation coaxial. The loss in kinetic energy of the system in the process is given by :

A

${{{I_1}{I_2}} \over {({I_1} + {I_2})}}{({\omega _1} - {\omega _2})^2}$

B

${{{{({I_1} - {I_2})}^2}{\omega _1}{\omega _2}} \over {2({I_1} + {I_2})}}$

C

${{{I_1}{I_2}} \over {2({I_1} + {I_2})}}{({\omega _1} - {\omega _2})^2}$

D

${{{{({\omega _1} - {\omega _2})}^2}} \over {2({I_1} + {I_2})}}$

Question 56: Palette 56/169

JEE Mains 2021 MCQ

Moment of inertia of a square plate of side l about the axis passing through one of the corner and perpendicular to the plane of square plate is given by :

A

${{M{l^2}} \over 6}$

B

${M{l^2}}$

C

${{M{l^2}} \over {12}}$

D

${2 \over 3}M{l^2}$

Question 57: Palette 57/169

JEE Mains 2021 MCQ

The solid cylinder of length 80 cm and mass M has a radius of 20 cm. Calculate the density of the material used if the moment of inertia of the cylinder about an axis CD parallel to AB as shown in figure is 2.7 kg m².

A

14.9 kg/m³

B

7.5 $\times$ 10¹ kg/m³

C

7.5 $\times$ 10² kg/m³

D

1.49 $\times$ 10² kg/m³

Question 58: Palette 58/169

JEE Mains 2021 MCQ

List-I	List-II
(a) MI of the rod (length L, Mass M, about an axis $ \bot $ to the rod passing through the midpoint)	(i) $8M{L^2}/3$
(b) MI of the rod (length L, Mass 2M, about an axis $ \bot $ to the rod passing through one of its end)	(ii) $M{L^2}/3$
(c) MI of the rod (length 2L, Mass M, about an axis $ \bot $ to the rod passing through its midpoint)	(iii) $M{L^2}/12$
(d) MI of the rod (Length 2L, Mass 2M, about an axis $ \bot $ to the rod passing through one of its end)	(iv) $2M{L^2}/3$

Choose the correct answer from the options given below:

A

(a)-(ii), (b)-(iii), (c)-(i), (d)-(iv)

B

(a)-(ii), (b)-(i), (c)-(iii), (d)-(iv)

C

(a)-(iii), (b)-(iv), (c)-(ii), (d)-(i)

D

(a)-(iii), (b)-(iv), (c)-(i), (d)-(ii)

Question 59: Palette 59/169

JEE Mains 2021 MCQ

The figure shows two solid discs with radius R and r respectively. If mass per unit area is same for both, what is the ratio of MI of bigger disc around axis AB (Which is $ \bot $ to the plane of the disc and passing through its centre) of MI of smaller disc around one of its diameters lying on its plane? Given 'M' is the mass of the larger disc. (MI stands for moment of inertia)

A

R² : r²

B

2r⁴ : R⁴

C

2R² : r²

D

2R⁴ : r⁴

Question 60: Palette 60/169

JEE Mains 2021 MCQ

Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R.

Assertion A : Moment of inertia of a circular disc of mass 'M' and radius 'R' about X, Y axes (passing through its plane) and Z-axis which is perpendicular to its plane were found to be I_x, I_y and I_z respectively. The respectively radii of gyration about all the three axes will be the same.

Reason R : A rigid body making rotational motion has fixed mass and shape. In the light of the above statements, choose the most appropriate answer from the options given below :

A

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

B

A is not correct but R is correct.

C

A is correct but R is not correct.

D

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

Question 61: Palette 61/169

JEE Mains 2021 MCQ

Consider a situation in which a ring, a solid cylinder and a solid sphere roll down on the same inclined plane without slipping. Assume that they start rolling from rest and having identical diameter.

The correct statement for this situation is

A

All of them will have same velocity.

B

The ring has greatest and the cylinder has the least velocity of the centre of mass at the bottom of the inclined plane.

C

The sphere has the greatest and the ring has the least velocity of the centre of mass at the bottom of the inclined plane.

D

The cylinder has the greatest and the sphere has the least velocity of the centre of mass at the bottom of the inclined plane.

Question 62: Palette 62/169

JEE Mains 2021 MCQ

A body rolls down an inclined plane without slipping. The kinetic energy of rotation is 50% of its translational kinetic energy. The body is :

A

Solid sphere

B

Solid cylinder

C

Hollow cylinder

D

Ring

Question 63: Palette 63/169

JEE Mains 2021 MCQ

Consider a uniform wire of mass M and length L. It is bent into a semicircle. Its moment of inertia about a line perpendicular to the plane of the wire passing through the center is :

A

${1 \over 4}{{M{L^2}} \over {{\pi ^2}}}$

B

${1 \over 2}{{M{L^2}} \over {{\pi ^2}}}$

C

${2 \over 5}{{M{L^2}} \over {{\pi ^2}}}$

D

${{M{L^2}} \over {{\pi ^2}}}$

Question 64: Palette 64/169

JEE Mains 2021 MCQ

A solid cylinder of mass m is wrapped with an inextensible light string and, is placed on a rough inclined plane as shown in the figure. The frictional force acting between the cylinder and the inclined plane is :


[The coefficient of static friction, $\mu$_s' is 0.4]

A

0

B

5 mg

C

${7 \over 2}$ mg

D

${{mg} \over 5}$

Question 65: Palette 65/169

JEE Mains 2021 MCQ

A thin circular ring of mass M and radius r is rotating about its axis with an angular speed $\omega$. Two particles having mass m each are now attached at diametrically opposite points. The angular speed of the ring will become :

A

$\omega {M \over {M + m}}$

B

$\omega {{M + 2m} \over M}$

C

$\omega {M \over {M + 2m}}$

D

$\omega {{M - 2m} \over {M + 2m}}$

Question 66: Palette 66/169

JEE Mains 2021 MCQ

A sphere of mass 2 kg and radius 0.5 m is rolling with an initial speed of 1 ms^-1 goes up an inclined plane which makes an angle of 30$^\circ$ with the horizontal plane, without slipping. How long will the sphere take to return to the starting point A?

A

0.60 s

B

0.52 s

C

0.80 s

D

0.57 s

Question 67: Palette 67/169

JEE Mains 2021 MCQ

A triangular plate is shown. A force $\overrightarrow F $ = 4$\widehat i$ $-$ 3$\widehat j$ is applied at point P. The torque at point P with respect to point 'O' and 'Q' are :

A

$-$ 15 + 20$\sqrt 3 $, 15 + 20$\sqrt 3 $

B

15 $-$ 20$\sqrt 3 $, 15 + 20$\sqrt 3 $

C

15 + 20$\sqrt 3 $, 15 $-$ 20$\sqrt 3 $

D

$-$ 15 $-$ 20$\sqrt 3 $, 15 $-$ 20$\sqrt 3 $

Question 68: Palette 68/169

JEE Mains 2021 MCQ

A mass M hangs on a massless rod of length l which rotates at a constant angular frequency. The mass M moves with steady speed in a circular path of constant radius. Assume that the system is in steady circular motion with constant angular velocity $\omega$. The angular momentum of M about point A is L_A which lies in the positive z direction and the angular momentum of M about point B is L_B. The correct statement for this system is :

A

L_A is constant, both in magnitude and direction

B

L_B is constant in direction with varying magnitude

C

L_B is constant, both in magnitude and direction

D

L_A and L_B are both constant in magnitude and direction

Question 69: Palette 69/169

JEE Mains 2021 MCQ

A cord is wound round the circumference of wheel of radius r. The axis of the wheel is horizontal and the moment of inertia about it is I. A weight mg is attached to the cord at the end. The weight falls from rest. After falling through a distance 'h', the square of angular velocity of wheel will be :

A

${{2mgh} \over {I + 2m{r^2}}}$

B

${{2mgh} \over {I + m{r^2}}}$

C

2gh

D

${{2gh} \over {I + m{r^2}}}$

Question 70: Palette 70/169

JEE Mains 2021 MCQ

Four identical solid spheres each of mass 'm' and radius 'a' are placed with their centres on the four corners of a square of side 'b'. The moment of inertia of the system about one side of square where the axis of rotation is parallel to the plane of the square is :

A

${4 \over 5}m{a^2}$

B

${8 \over 5}m{a^2} + m{b^2}$

C

${4 \over 5}m{a^2} + 2m{b^2}$

D

${8 \over 5}m{a^2} + 2m{b^2}$

Question 71: Palette 71/169

JEE Mains 2021 MCQ

A sphere of radius 'a' and mass 'm' rolls along a horizontal plane with constant speed v₀. It encounters an inclined plane at angle $\theta$ and climbs upward. Assuming that it rolls without slipping, how far up the sphere will travel?

A

${{v_0^2} \over {2g\sin \theta }}$

B

${{7v_0^2} \over {10g\sin \theta }}$

C

${2 \over 5}{{v_0^2} \over {g\sin \theta }}$

D

${{v_0^2} \over {5g\sin \theta }}$

Question 72: Palette 72/169

JEE Mains 2021 MCQ

Moment of inertia (M. I.) of four bodies, having same mass and radius, are reported as;

I₁ = M.I. of thin circular ring about its diameter,

I₂ = M.I. of circular disc about an axis perpendicular to disc and going through the centre,

I₃ = M.I. of solid cylinder about its axis and

I₄ = M.I. of solid sphere about its diameter.

Then :

A

I₁ = I₂ = I₃ > I₄

B

I₁ + I₃ < I₂ + I₄

C

I₁ = I₂ = I₃ < I₄

D

I₁ + I₂ = I₃ + ${5 \over 2}$ I₄

Question 73: Palette 73/169

JEE Mains 2020 MCQ

The linear mass density of a thin rod AB of length L varies from A to B as
$\lambda \left( x \right) = {\lambda _0}\left( {1 + {x \over L}} \right)$, where x is the distance from A. If M is the mass of the rod then its moment of inertia about an axis passing through A and perpendicular to the rod is :

A

${2 \over 5}M{L^2}$

B

${5 \over {12}}M{L^2}$

C

${7 \over {18}}M{L^2}$

D

${3 \over 7}M{L^2}$

Question 74: Palette 74/169

JEE Mains 2020 MCQ

Four point masses, each of mass m, are fixed at the corners of a square of side $l$. The square is rotating with angular frequency $\omega $, about an axis passing through one of the corners of the square and parallel to its diagonal, as shown in the figure. The angular momentum of the square about this axis is :

A

3m$l$²$\omega $

B

4m$l$²$\omega $

C

m$l$²$\omega $

D

2m$l$²$\omega $

Question 75: Palette 75/169

JEE Mains 2020 MCQ

Shown in the figure is a hollow icecream cone (it is open at the top). If its mass is M, radius of its top, R and height, H, then its moment of inertia about its axis is :

A

${{M\left( {{R^2} + {H^2}} \right)} \over 3}$

B

${{M{R^2}} \over 2}$

C

${{M{R^2}} \over 3}$

D

${{M{H^2}} \over 3}$

Question 76: Palette 76/169

JEE Mains 2020 MCQ

A ring is hung on a nail. It can oscillate, without slipping or sliding
(i) in its plane with a time period T₁ and,
(ii) back and forth in a direction perpendicular to its plane,
with a period T₂. The ratio ${{{T_1}} \over {{T_2}}}$ will be :

A

${{\sqrt 2 } \over 3}$

B

${2 \over {\sqrt 3 }}$

C

${2 \over 3}$

D

${3 \over {\sqrt 2 }}$

Question 77: Palette 77/169

JEE Mains 2020 MCQ

A wheel is rotating freely with an angular speed $\omega $ on a shaft. The moment of inertia of the wheel is I and the moment of inertia of the shaft is negligible. Another wheel of moment of inertia 3I initially at rest is suddenly coupled to the same shaft. The resultant fractional loss in the kinetic energy of the system is :

A

0

B

${5 \over 6}$

C

${1 \over 4}$

D

${3 \over 4}$

Question 78: Palette 78/169

JEE Mains 2020 MCQ

For a uniform rectangular sheet shown in the figure, the ratio of moments of inertia about the axes perpendicular to the sheet and passing through O (the centre of mass) and O' (corner point) is :

A

${1 \over 2}$

B

${1 \over 4}$

C

${1 \over 8}$

D

${2 \over 3}$

Question 79: Palette 79/169

JEE Mains 2020 MCQ

Consider two uniform discs of the same thickness and different radii R₁ = R and
R₂ = $\alpha $R made of the same material. If the ratio of their moments of inertia I₁ and I₂ , respectively, about their axes is I₁ : I₂ = 1 : 16 then the value of $\alpha $ is :

A

$\sqrt 2 $

B

2

C

$2\sqrt 2 $

D

4

Question 80: Palette 80/169

JEE Mains 2020 MCQ

A uniform rod of length ‘$l$’ is pivoted at one of its ends on a vertical shaft of negligible radius. When the shaft rotates at angular speed $\omega $ the rod makes an angle $\theta $ with it (see figure). To find $\theta $ equate the rate of change of angular momentum (direction going into the paper) ${{m{l^2}} \over {12}}{\omega ^2}\sin \theta \cos \theta $ about the centre of mass (CM) to the torque provided by the horizontal and vertical forces F_H and F_V about the CM. The value of $\theta $ is then such that :

A

$\cos \theta = {{2g} \over {3l{\omega ^2}}}$

B

$\cos \theta = {{3g} \over {2l{\omega ^2}}}$

C

$\cos \theta = {g \over {2l{\omega ^2}}}$

D

$\cos \theta = {g \over {l{\omega ^2}}}$

Question 81: Palette 81/169

JEE Mains 2020 MCQ

Moment of inertia of a cylinder of mass M, length L and radius R about an axis passing through its centre and perpendicular to the axis of the cylinder is
I = $M\left( {{{{R^2}} \over 4} + {{{L^2}} \over {12}}} \right)$. If such a cylinder is to be made for a given mass of a material, the ratio ${L \over R}$ for it to have minimum possible I is

A

${3 \over 2}$

B

$\sqrt {{3 \over 2}} $

C

$\sqrt {{2 \over 3}} $

D

${{2 \over 3}}$

Question 82: Palette 82/169

JEE Mains 2020 MCQ

Two uniform circular discs are rotating independently in the same direction around their common axis passing through their centres. The moment of inertia and angular velocity of the first disc are 0.1 kg-m² and 10 rad s^–1 respectively while those for the second one are 0.2 kg-m² and 5 rad s^–1 respectively. At some instant they get stuck together and start rotating as a single system about their common axis with some angular speed. The kinetic energy of the combined system is :

A

${{20} \over 3}J$

B

${{5} \over 3}J$

C

${{10} \over 3}J$

D

${{2} \over 3}J$

Question 83: Palette 83/169

JEE Mains 2020 MCQ

A uniform cylinder of mass M and radius R is to be pulled over a step of height a (a < R) by applying a force F at its centre ‘O’ perpendicular to the plane through the axes of the cylinder on the edge of the step (see figure). The minimum value of F required is :

A

$Mg\sqrt {1 - {{\left( {{{R - a} \over R}} \right)}^2}} $

B

$Mg\sqrt {1 - {{{a^2}} \over {{R^2}}}} $

C

$Mg{a \over R}$

D

$Mg\sqrt {{{\left( {{R \over {R - a}}} \right)}^2} - 1} $

Question 84: Palette 84/169

JEE Mains 2020 MCQ

Shown in the figure is rigid and uniform one meter long rod AB held in horizontal position by two strings tied to its ends and attached to the ceiling. The rod is of mass ‘m’ and has another weight of mass 2 m hung at a distance of 75 cm from A. The tension in the string at A is :

A

0.5 mg

B

2 mg

C

0.75 mg

D

1 mg

Question 85: Palette 85/169

JEE Mains 2020 MCQ

A uniformly thick wheel with moment of inertia I and radius R is free to rotate about its centre of mass (see fig). A massless string is wrapped over its rim and two blocks of masses m₁ and m₂ (m₁ $ > $ m₂) are attached to the ends of the string. The system is released from rest. The angular speed of the wheel when m₁ descents by a distance h is :

A

${\left[ {{{2\left( {{m_1} + {m_2}} \right)gh} \over {\left( {{m_1} + {m_2}} \right){R^2} + I}}} \right]^{{1 \over 2}}}$

B

${\left[ {{{{m_1} + {m_2}} \over {\left( {{m_1} + {m_2}} \right){R^2} + I}}} \right]^{{1 \over 2}}}gh$

C

${\left[ {{{\left( {{m_1} - {m_2}} \right)} \over {\left( {{m_1} + {m_2}} \right){R^2} + I}}} \right]^{{1 \over 2}}}gh$

D

${\left[ {{{2\left( {{m_1} - {m_2}} \right)gh} \over {\left( {{m_1} + {m_2}} \right){R^2} + I}}} \right]^{{1 \over 2}}}$

Question 86: Palette 86/169

JEE Mains 2020 MCQ

Three solid spheres each of mass m and diameter d are stuck together such that the lines connecting the centres form an equilateral triangle of side of length d. The ratio I₀/I_A of moment of inertia I₀ of the system about an axis passing the centroid and about center of any of the spheres I_A and perpendicular to the plane of the triangle is :

A

${{13} \over {23}}$

B

${{23} \over {13}}$

C

${{15} \over {13}}$

D

${{13} \over {15}}$

Question 87: Palette 87/169

JEE Mains 2020 MCQ

A uniform sphere of mass 500 g rolls without slipping on a plane horizontal surface with its centre moving at a speed of 5.00 cm/s. Its kinetic energy is :

A

8.75 × 10^–3 J

B

1.13 × 10^–3 J

C

8.75 × 10^–4 J

D

6.25 × 10^–4 J

Question 88: Palette 88/169

JEE Mains 2020 MCQ

Consider a uniform rod of mass M = 4m and length $\ell $ pivoted about its centre. A mass m moving with velocity v making angle $\theta = {\pi \over 4}$ to the rod's long axis collides with one end of the rod and sticks to it. The angular speed of the rod-mass system just after the collision is :

A

${{3\sqrt 2 } \over 7}{v \over \ell }$

B

${3 \over 7}{v \over \ell }$

C

${3 \over {7\sqrt 2 }}{v \over \ell }$

D

${4 \over 7}{v \over \ell }$

Question 89: Palette 89/169

JEE Mains 2020 MCQ

Mass per unit area of a circular disc of radius $a$ depends on the distance r from its centre as $\sigma \left( r \right)$ = A + Br . The moment of inertia of the disc about the axis, perpendicular to the plane and assing through its centre is:

A

$2\pi {a^4}\left( {{A \over 4} + {{aB} \over 5}} \right)$

B

$\pi {a^4}\left( {{A \over 4} + {{aB} \over 5}} \right)$

C

$2\pi {a^4}\left( {{{aA} \over 4} + {B \over 5}} \right)$

D

$2\pi {a^4}\left( {{A \over 4} + {B \over 5}} \right)$

Question 90: Palette 90/169

JEE Mains 2020 MCQ

The radius of gyration of a uniform rod of length $l$, about an axis passing through a point ${l \over 4}$ away from the centre of the rod, and perpendicular to it, is :

A

${1 \over 8}l$

B

${1 \over 4}l$

C

$\sqrt {{7 \over {48}}} l$

D

$\sqrt {{3 \over 8}} l$

Question 91: Palette 91/169

JEE Mains 2020 MCQ

As shown in the figure, a bob of mass m is tied by a massless string whose other end portion is wound on a fly wheel (disc) of radius r and mass m. When released from rest the bob starts falling vertically. When it has covered a distance of h, the angular speed of the wheel will be:

A

$r\sqrt {{3 \over {2gh}}} $

B

$r\sqrt {{3 \over {4gh}}} $

C

${1 \over r}\sqrt {{{4gh} \over 3}} $

D

${1 \over r}\sqrt {{{2gh} \over 3}} $

Question 92: Palette 92/169

JEE Mains 2019 MCQ

A uniform rod of length $\ell $ is being rotated in a horizontal plane with a constant angular speed about an axis passing through one of its ends. If the tension generated in the rod due to rotation is T(x) at a distance x from the axis, then which of the following graphs depicts it most closely?

A

B

C

D

Question 93: Palette 93/169

JEE Mains 2019 MCQ

A circular disc of radius b has a hole of radius a at its centre (see figure). If the mass per unit area of the disc varies as $\left( {{{{\sigma _0}} \over r}} \right)$ , then the radius of gyration of the disc about its axis passing through the centre is:

A

$\sqrt {{{{a^2} + {b^2} + ab} \over 2}} $

B

$\sqrt {{{a + b} \over 3}} $

C

$\sqrt {{{{a^2} + {b^2} + ab} \over 3}} $

D

$\sqrt {{{a + b} \over 2}} $

Question 94: Palette 94/169

JEE Mains 2019 MCQ

A person of mass M is, sitting on a swing of length L and swinging with an angular amplitude $\theta $₀. If the person stands up when the swing passes through its lowest point, the work done by him, assuming that his center of mass moves by a distance $\ell $($\ell $ << L), is close to;

A

mg$\ell $(1 + $\theta $₀²)

B

mg$\ell $

C

mg$\ell $(1 + ${{\theta _0^2} \over 2}$)

D

mg$\ell $(1 - $\theta $₀²)

Question 95: Palette 95/169

JEE Mains 2019 MCQ

The time dependence of the position of a particle of mass m = 2 is given by $\overrightarrow r \left( t \right) = 2t\widehat i - 3{t^2}\widehat j$ . Its angular momentum, with respect to the origin, at time t = 2 is

A

36 $\widehat k$

B

- 48 $\widehat k$

C

$ - 34\left( {\widehat k - \widehat i} \right)$

D

$48\left( {\widehat i + \widehat j} \right)$

Question 96: Palette 96/169

JEE Mains 2019 MCQ

A metal coin of mass 5 g and radius 1 cm is fixed to a thin stick AB of negligible mass as shown in the figure. The system is initially at rest. The constant torque, that will make the system rotate about AB at 25 rotations per second in 5s, is close to :

A

7.9 × 10^–6 Nm

B

4.0 × 10^–6 Nm

C

2.0 × 10^–5 Nm

D

1.6 × 10^–5 Nm

Question 97: Palette 97/169

JEE Mains 2019 MCQ

A solid sphere of mass M and radius R is divided into two unequal parts. The first part has a mass of ${{7M} \over 8}$ and is converted into a uniform disc of radius 2R. The second part is converted into a uniform solid sphere. Let I₁ be the moment of inertia of the disc about its axis and I₂ be the moment of inertia of the new sphere about its axis. The ratio I₁/I₂ is given by :

A

65

B

140

C

185

D

285

Question 98: Palette 98/169

JEE Mains 2019 MCQ

Two coaxial discs, having moments of inertia I₁ and I₁/2, are rotating with respective angular velocities $\omega $₁ and $\omega $₁/2 , about their common axis. They are brought in contact with each other and thereafter they rotate with a common angular velocity. If E_f and E_i are the final and initial total energies, then (E_f - E_i) is:

A

${{{I_1}\omega _1^2} \over {24}}$

B

${{{I_1}\omega _1^2} \over {12}}$

C

${3 \over 8}{I_1}\omega _1^2$

D

${{{I_1}\omega _1^2} \over {6}}$

Question 99: Palette 99/169

JEE Mains 2019 MCQ

A particle of mass m is moving along a trajectory given by
x = x₀ + a cos$\omega $₁t
y = y₀ + b sin$\omega $₂t
The torque, acting on the particle about the origin, at t = 0 is :

A

Zero

B

+my₀a $\omega _1^2$$\widehat k$

C

$ - m\left( {{x_0}b\omega _2^2 - {y_0}a\omega _1^2} \right)\widehat k$

D

m (–x₀b + y₀a) $\omega _1^2$$\widehat k$

Question 100: Palette 100/169

JEE Mains 2019 MCQ

A thin disc of mass M and radius R has mass per unit area $\sigma $(r) = kr² where r is the distance from its centre. Its moment of inertia about an axis going through its centre of mass and perpendicular to its plane is :

A

${{M{R^2}} \over 3}$

B

${{M{R^2}} \over 6}$

C

${{2M{R^2}} \over 3}$

D

${{M{R^2}} \over 2}$

Question 101: Palette 101/169

JEE Mains 2019 MCQ

A thin smooth rod of length L and mass M is rotating freely with angular speed $\omega $₀ about an axis perpendicular to the rod and passing through its center. Two beads of mass m and negligible size are at the center of the rod initially. The beads are free to slide along the rod. The angular speed of the system , when the beads reach the opposite ends of the rod, will be :-

A

${{M{\omega _0}} \over {M + 3m}}$

B

${{M{\omega _0}} \over {M + m}}$

C

${{M{\omega _0}} \over {M + 6m}}$

D

${{M{\omega _0}} \over {M + 2m}}$

Question 102: Palette 102/169

JEE Mains 2019 MCQ

Moment of inertia of a body about a given axis is 1.5 kg m². Initially the body is at rest. In order to produce a rotational kinetic energy of 1200 J, the angular accleration of 20 rad/s² must be applied about the axis for a duration of :-

A

2.5 s

B

3 s

C

5s

D

2 s

Question 103: Palette 103/169

JEE Mains 2019 MCQ

The following bodies are made to roll up (without slipping) the same inclined plane from a horizontal plane. : (i) a ring of radius R, (ii) a solid cylinder of radius R/2 and (iii) a solid sphere of radius R/4 . If in each case, the speed of the centre of mass at the bottom of the incline is same, the ratio of the maximum heights they climb is :

A

20 : 15 : 14

B

4 : 3 : 2

C

2 : 3 : 4

D

10 : 15 : 7

Question 104: Palette 104/169

JEE Mains 2019 MCQ

A stationary horizontal disc is free to rotate about its axis. When a torque is applied on it, its kinetic energy as a function of $\theta $, where $\theta $ is the angle by which it has rotated, is given as k$\theta $². If its moment of inertia is I then the angular acceleration of the disc is :

A

${k \over {4I}}\theta $

B

${k \over {I}}\theta $

C

${k \over {2I}}\theta $

D

${2k \over {I}}\theta $

Question 105: Palette 105/169

JEE Mains 2019 MCQ

A rectangular solid box of length 0.3 m is held horizontally, with one of its sides on the edge of a platform of height 5m. When released, it slips off the table in a very short time t = 0.01s, remaining essentially horizontal. The angle by which it would rotate when it hits the ground will be (in radians) close to :-

A

0.28

B

0.02

C

0.3

D

0.5

Question 106: Palette 106/169

JEE Mains 2019 MCQ

A solid sphere and solid cylinder of identical radii approach an incline with the same linear velocity (see figure). Both roll without slipping all throughout. The two climb maximum heights h_sph and h_cyl on the incline. The ratio h_sph/h_cyl is given by :-

A

1

B

14/15

C

4/5

D

2/$\sqrt5$

Question 107: Palette 107/169

JEE Mains 2019 MCQ

An electric dipole is formed by two equal and opposite charges q with separation d. The charges have same mass m. It is kept in a uniform electric field E. If it is slightly rotated from its equilibrium orientation, then its angular frequency $\omega$ is :-

A

$\sqrt {{{qE} \over {md}}} $

B

$\sqrt {{{qE} \over {2md}}} $

C

$\sqrt {{{qE} \over {-2md}}} $

D

$\sqrt {{{2qE} \over {md}}} $

Question 108: Palette 108/169

JEE Mains 2019 MCQ

A thin circular plate of mass M and radius R has its density varying as $\rho $(r) = $\rho $₀r with $\rho $₀ as constant and r is the distance from its centre. The moment of Inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is I = aMR². The value of the coefficient a is :

A

${1 \over 2}$

B

${3 \over 2}$

C

${8 \over 5}$

D

${3 \over 5}$

Question 109: Palette 109/169

JEE Mains 2019 MCQ

Two particles A, B are moving on two concentric circles of radii R₁ and R₂ with equal angular speed $\omega $. At t = 0, their positions and direction of motion are shown in the figure. :



The relative velocity ${\overrightarrow V _A} - {\overrightarrow V _B}$ at t = ${\pi \over {2\omega }}$ is given by :

A

$ - \omega \left( {{R_1} + {R_2}} \right)\,\widehat i$

B

$\omega \left( {{R_2} - {R_1}} \right)\,\widehat i$

C

$\omega \left( {{R_1} + {R_2}} \right)\,\widehat i$

D

$\omega \left( {{R_1} - {R_1}} \right)\,\widehat i$

Question 110: Palette 110/169

JEE Mains 2019 MCQ

A particle of mass 20 g is released with an initial velocity 5 m/s along the curve from the point A, as shown in the figure. The point A is at height h from point B. The particle slides along the frictionless surface. when the particle reaches point b, its angular momentum about O will be :
(Take g = 10 m/s²)

A

6 kg-m²/s

B

8 kg-m²/s

C

2 kg-m²/s

D

3 kg-m²/s

Question 111: Palette 111/169

JEE Mains 2019 MCQ

The moment of inertia of a solid sphere, about an axis parallel to its diameter and at a distance of x from it, is 'I(x)'. Which one of the graphs represents the variation of I(x) with x correctly ?

A

B

C

D

Question 112: Palette 112/169

JEE Mains 2019 MCQ

Let the moment of inertia of a hollow cylinder of length 30 cm (inner radius 10 cm and outer radius 20 cm), about its axis be I. The radius of a thin cylinder of the same mass such that its moment of inertia about its axis is also I, is :

A

16 cm

B

12 cm

C

14 cm

D

18 cm

Question 113: Palette 113/169

JEE Mains 2019 MCQ

A circular disc D₁ of mass M and radius R has two identical discs D₂ and D₃ of the same mass M and radius R attached rigidly at its opposite ends (see figure). The moment of inertia of the system about the axis OO' ,passing through the centre of D₁ as shown in the figure, will be :

A

3MR²

B

MR²

C

${2 \over 3}$ MR²

D

${4 \over 5}$ MR²

Question 114: Palette 114/169

JEE Mains 2019 MCQ

A string is wound around a hollow cylinder of mass 5 kg and radius 0.5m. If the string is now pulled with a horizontal force of 40 N, and the cylinder is rolling without slipping on a horizontal surface (see figure), then the angular acceleration of the cylinder will be (Neglect the mass and thickness of the string) :

A

16 rad/s²

B

20 rad/s²

C

10 rad/s²

D

12 rad/s²

Question 115: Palette 115/169

JEE Mains 2019 MCQ

The magnitude of torque on a particle of mass 1 kg is 2.5 Nm about the origin. If the force acting on it is 1 N, and the distance of the particle from the origin is 5m, the angle between the force and the position vector is (in radians) :

A

${\pi \over 8}$

B

${\pi \over 6}$

C

${\pi \over 4}$

D

${\pi \over 3}$

Question 116: Palette 116/169

JEE Mains 2019 MCQ

A slab is subjected to two forces $\overrightarrow {{F_1}} $ and $\overrightarrow {{F_2}} $ of same magnitude F as shown in the figure. Force $\overrightarrow {{F_2}} $ is in XY-plane while force $\overrightarrow {{F_1}} $ acts along z = axis at the point $\left( {2\overrightarrow i + 3\overrightarrow j } \right).$. The moment of these forces about point O will be :

A

$\left( {3\widehat i - 2\widehat j - 3\widehat k} \right)F$

B

$\left( {3\widehat i + 2\widehat j - 3\widehat k} \right)F$

C

$\left( {3\widehat i + 2\widehat j + 3\widehat k} \right)F$

D

$\left( {3\widehat i - 2\widehat j + 3\widehat k} \right)F$

Question 117: Palette 117/169

JEE Mains 2019 MCQ

An equilateral triangle ABC is cut from a thin solid sheet of wood. (see figure) D, E and F are the mid-points of its sides as shown and G is the centre of the triangle. The moment of inertia of the triangle about an axis passing through G and perpendicular to the plane of the triangle is I0. If the smaller triangle DEF is removed from ABC, the moment of inertia of the remaining figure about the same axis is I. then :

A

${\rm I} = {{{{\rm I}_0}} \over 4}$

B

${\rm I} = {{15} \over {16}}{{\rm I}_0}$

C

${\rm I} = {9 \over {16}}{{\rm I}_0}$

D

${\rm I} = {3 \over 4}{{\rm I}_0}$

Question 118: Palette 118/169

JEE Mains 2019 MCQ

A rigid massless rod of length 3l has two masses attached at each end as shown in the figure. The rod is pivoted at point P on the horizontal axis (see figure). When released from initial horizontal position, its instantaneous angular acceleration will be -

A

${g \over {13l}}$

B

${g \over {2l}}$

C

${g \over {3l}}$

D

${7g \over {3l}}$

Question 119: Palette 119/169

JEE Mains 2019 MCQ

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 :

A

${{17} \over {15}}$ MR²

B

${{137} \over {15}}$ MR²

C

${{209} \over {15}}$ MR²

D

${{152} \over {15}}$ MR²

Question 120: Palette 120/169

JEE Mains 2019 MCQ

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 -

A

$\mu $FR/2

B

$\mu $FR/3

C

$\mu $FR/6

D

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

Question 121: Palette 121/169

JEE Mains 2019 MCQ

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 -

A

${F \over {2mR}}$

B

${2F \over {3mR}}$

C

${3F \over {2mR}}$

D

${F \over {3mR}}$

Question 122: Palette 122/169

JEE Mains 2019 MCQ

A rod of length 50 cm is pivoted at one end. It is raised such that if makes an angle of 30^o 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)

A

$\sqrt {{{30} \over 7}} $

B

$\sqrt {30} $

C

${{\sqrt {20} } \over 3}$

D

${{\sqrt {30} } \over 2}$

Question 123: Palette 123/169

JEE Mains 2019 MCQ

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 :

A

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

B

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

C

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

D

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

Question 124: Palette 124/169

JEE Mains 2019 MCQ

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 :

A

${L \over m}$

B

${4L \over m}$

C

${L \over 2m}$

D

${2L \over m}$

Question 125: Palette 125/169

JEE Mains 2018 MCQ

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 :

A

1 : 3

B

1 : 4

C

1 : 5

D

1 : 2

Question 126: Palette 126/169

JEE Mains 2018 MCQ

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 :

A

${{37} \over 9}M{R^2}$

B

$4M{R^2}$

C

${{40} \over 9}M{R^2}$

D

$10M{R^2}$

Question 127: Palette 127/169

JEE Mains 2018 MCQ

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 :

A

${{181} \over 2}M{R^2}$

B

${{55} \over 2}M{R^2}$

C

${{19} \over 2}M{R^2}$

D

${{73} \over 2}M{R^2}$

Question 128: Palette 128/169

JEE Mains 2018 MCQ

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 :

A

${v \over {5L}}$

B

${6v \over {5L}}$

C

${3v \over {5L}}$

D

${v \over {6L}}$

Question 129: Palette 129/169

JEE Mains 2018 MCQ

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)

A

$\sqrt {\sin \alpha } $

B

${\sin \alpha }$

C

$\sqrt {\cos \alpha } $

D

${\cos \alpha }$

Question 130: Palette 130/169

JEE Mains 2018 MCQ

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 :

A

$m,x$

B

$m,{1 \over x}$

C

$m,{1 \over {{x^2}}}$

D

$m,{x^2}$

Question 131: Palette 131/169

JEE Mains 2018 MCQ

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 :

A

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

B

$\tan \theta = 2$

C

$\tan \theta = 4$

D

$\tan \theta = {1 \over 4}$

Question 132: Palette 132/169

JEE Mains 2017 MCQ

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 :

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

Question 133: Palette 133/169

JEE Mains 2017 MCQ

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 :

A

${{219\,M{R^2}} \over {256}}$

B

${{237\,M{R^2}} \over {512}}$

C

${{19\,M{R^2}} \over {512}}$

D

${{197\,M{R^2}} \over {256}}$

Question 134: Palette 134/169

JEE Mains 2017 MCQ

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 :

A

${{2\,\,mg} \over {2\,m + M}}$

B

${{2\,\,Mg} \over {2\,m + M}}$

C

${{2\,\,mg} \over {2\,M + m}}$

D

${{2\,\,Mg} \over {2\,M + M}}$

Question 135: Palette 135/169

JEE Mains 2017 MCQ

Moment of inertia of an equilateral triangular lamina ABC, about the axis passing through its centre O and perpendicular to its plane is I_o 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 :

A

${7 \over 8}$ I_o

B

${15 \over 16}$ I_o

C

${{3\,{{\rm I}_o}} \over 4}$

D

${{31\,{{\rm I}_o}} \over 32}$

Question 136: Palette 136/169

JEE Mains 2017 MCQ

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 ?

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.

Question 137: Palette 137/169

JEE Mains 2017 MCQ

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

A

${{2g} \over {3l}}\cos \theta $

B

${{3g} \over {2l}}\sin \theta $

C

${{2g} \over {3l}}\sin \theta $

D

${{3g} \over {3l}}\sin \theta $

Question 138: Palette 138/169

JEE Mains 2017 MCQ

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?

A

${3 \over {\sqrt 2 }}$

B

$\sqrt {{3 \over 2}} $

C

${{\sqrt 3 } \over 2}$

D

1

Question 139: Palette 139/169

JEE Mains 2016 MCQ

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) :

A

0.4

B

1.3

C

8.0

D

27.0

Question 140: Palette 140/169

JEE Mains 2016 MCQ

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 :

A

5.0

B

6.7

C

9.4

D

13.3

Question 141: Palette 141/169

JEE Mains 2016 MCQ

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 :

A

go straight

B

turn left and right alternately

C

turn left

D

turn right

Question 142: Palette 142/169

JEE Mains 2016 MSQ

A particle of mass m is moving along the side of a square of side ‘a’, with a uniform speed v in the x-y plane as shown in the figure :


Which of the following statements is false for the angular momentum $\overrightarrow L $ about the origin ?

A

$\overrightarrow L = mv\left[ {{R \over {\sqrt 2 }} + a} \right]\widehat k$
when the particle is moving from B to C.

B

$\overrightarrow L = {{mv} \over {\sqrt 2 }}R\widehat k$
when the particle is moving from D to A.

C

$\overrightarrow L = - {{mv} \over {\sqrt 2 }}R\widehat k$
when the particle is moving from A to B

D

$\overrightarrow L = mv\left[ {{R \over {\sqrt 2 }} - a} \right]\widehat k$
when the particle is moving from C to D.

Question 143: Palette 143/169

JEE Mains 2015 MCQ

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:

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

${{M{R^2}} \over {16\sqrt {2\pi } }}$

Question 144: Palette 144/169

JEE Mains 2014 MCQ

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:

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.

Question 145: Palette 145/169

JEE Mains 2014 MCQ

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?

A

${{2g} \over 3}$

B

${{g} \over 2}$

C

${{5g} \over 6}$

D

$g$

Question 146: Palette 146/169

JEE Mains 2013 MCQ

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?

A

${{r{\omega _0}} \over 4}$

B

${{r{\omega _0}} \over 3}$

C

${{r{\omega _0}} \over 2}$

D

${r{\omega _0}}$

Question 147: Palette 147/169

JEE Mains 2011 MCQ

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:

A

more than $3$ but less than $6$

B

more than $6$ but less than $9$

C

more than $9$

D

less than $3$

Question 148: Palette 148/169

JEE Mains 2011 MCQ

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:

A

$g$

B

${2 \over 3}g$

C

${g \over 3}$

D

${3 \over 2}g$

Question 149: Palette 149/169

JEE Mains 2011 MCQ

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.

A

continuously decreases

B

continuously increases

C

first increases and then decreases

D

remains unchanged

Question 150: Palette 150/169

JEE Mains 2010 MCQ

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.

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

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

Question 151: Palette 151/169

JEE Mains 2009 MCQ

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:

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

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

Question 152: Palette 152/169

JEE Mains 2008 MCQ

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

A

${5 \over 6}m{a^2}$

B

${1 \over 12}m{a^2}$

C

${7 \over 12}m{a^2}$

D

${2 \over 3}m{a^2}$

Question 153: Palette 153/169

JEE Mains 2007 MCQ

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

A

${I_{AC}} = \sqrt 2 \,\,{I_{EF}}$

B

$\sqrt 2 {I_{AC}} = {I_{EF}}$

C

${I_{AD}} = 3{I_{EF}}$

D

${I_{AC}} = {I_{EF}}$

Question 154: Palette 154/169

JEE Mains 2007 MCQ

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

A

constant torque

B

constant force

C

constant linear momentum

D

zero torque

Question 155: Palette 155/169

JEE Mains 2007 MCQ

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

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

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

Question 156: Palette 156/169

JEE Mains 2006 MCQ

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

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)$

Question 157: Palette 157/169

JEE Mains 2006 MCQ

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

A

$2m{l^2}$

B

$\sqrt 3 m{l^2}$

C

$3m{l^2}$

D

$m{l^2}$

Question 158: Palette 158/169

JEE Mains 2006 MCQ

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 ' = $

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

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

Question 159: Palette 159/169

JEE Mains 2005 MCQ

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

A

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

B

${{{{R_2}} \over {{R_1}}}}$

C

${{{{R_1}} \over {{R_2}}}}$

D

$1$

Question 160: Palette 160/169

JEE Mains 2005 MCQ

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

A

${2 \over 5}M{r^2}$

B

${1 \over 4}Mr$

C

${1 \over 2}M{r^2}$

D

$M{r^2}$

Question 161: Palette 161/169

JEE Mains 2004 MCQ

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

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.

Question 162: Palette 162/169

JEE Mains 2004 MCQ

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 ?

A

Angular velocity

B

Angular momentum

C

Moment of inertia

D

Rotational kinetic energy

Question 163: Palette 163/169

JEE Mains 2003 MCQ

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

A

${I_Y} = 32{I_X}$

B

${I_Y} = 16{I_X}$

C

${I_Y} = {I_X}$

D

${I_Y} = 64{I_X}$

Question 164: Palette 164/169

JEE Mains 2003 MCQ

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

A

${L \over 4}$

B

$2L$

C

$4L$

D

${L \over 2}$

Question 165: Palette 165/169

JEE Mains 2003 MCQ

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

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 $\overrightarrow {F.} \overrightarrow \tau = 0\,\,$

Question 166: Palette 166/169

JEE Mains 2002 MCQ

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

A

${{M{R^2}} \over 2}$

B

$M{R^2}$

C

$2M{R^2}$

D

${{M{R^2}} \over 4}$

Question 167: Palette 167/169

JEE Mains 2002 MCQ

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)

A

solid sphere

B

hollow sphere

C

ring

D

all same

Question 168: Palette 168/169

JEE Mains 2002 MCQ

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

A

$mvL$

B

$mvl$

C

$mvr$

D

zero

Question 169: Palette 169/169

JEE Mains 2002 MCQ

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?

A

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

B

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

C

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

D

$\left( {{M \over {M + 2m}}} \right)\,\,{\omega _1}$