How many questions from Rotational Motion appear in JEE?

Rotational Motion has appeared in 172 of the last 186 JEE Main and JEE Advanced papers — about 92% of them — contributing 266 questions in total across those papers.

Is Rotational Motion an important chapter for JEE?

Judged by how often it is actually tested, it appears in roughly 92% of papers. Chapters above about 50% are effectively guaranteed to show up every session, so they repay thorough preparation; lower-frequency chapters are better treated as targeted revision.

Where do these Rotational Motion questions come from?

Every question is from an official JEE Main or JEE Advanced paper, transcribed from the original paper and tagged to this chapter. Answers follow the official answer key.

Rotational Motion JEE Previous Year Questions with Answers

Q1·PhysicsSingle correctJEE Advanced 2026

Consider a large disk of radius

R

and two smaller disks, each of radius

r = R/50

, lying on its circumference, as shown in the figure. The smaller disks are initially in contact with each other, with an angular separation

\Delta\theta

between their centers. They are made to roll without slipping in opposite directions, with constant angular velocities

\omega

and

2\omega

while the large disk is held stationary. The time

\tau

at which the smaller disks are again in contact is: [Use

\sin(\Delta\theta) = \Delta\theta

and ignore gravity.]

(A) $\tau = 51 \times \left(2\pi - \frac{4}{51}\right)/\omega$
(B) $\tau = 51 \times \left(2\pi - \frac{2}{51}\right)/3\omega$
(C) $\tau = 51 \times \left(2\pi - \frac{4}{51}\right)/3\omega$
(D) $\tau = 51 \times \left(2\pi - \frac{2}{51}\right)/\omega$

Q2·PhysicsNumericalJEE Advanced 2026

Passage: A uniform circular disk of radius 0.2 m and mass 1 kg is pivoted at its top point

C

such that it can rotate freely around

C

in the

XY

plane, as shown in the figure. Initially, when the disk is at rest, a particle of mass 20 g, travelling along negative

x

direction in the

XY

plane with speed 100 ms

^{-1}

, hits the circumference of the disk at a point

P

. After collision the particle moves along negative

y

direction at a speed of 90 ms

^{-1}

. [Given: the acceleration due to gravity

(g) = -10\hat{j}

ms

^{-2}

] Question: After the collision the disk starts to rotate around point

C

in the

XY

plane. The maximum change in the height (in m) of its center

O

is:

Q3·PhysicsSingle correctJEE Advanced 2026

List-I shows four planar structures made of uniform solid rods each of mass

m

and length

l

. In the List-II the possible moment of inertia of these structures about an axis

OCO'

, which lies in the plane of the structures, are given. Choose the option that describes the correct match between the entries in List-I to those in List-II.

List-I	List-II
P.see figure	1. $\frac{5}{4}ml^2$
Q.see figure	2. $\frac{1}{6}ml^2$
R.see figure	3. $\frac{1}{12}ml^2$
S.see figure	4. $\frac{2}{3}ml^2$
	5. $\frac{1}{3}ml^2$

(A)P→ 5, Q→ 1, R→ 4, S→ 2
(B)P→ 1, Q→ 3, R→ 4, S→ 2
(C)P→ 5, Q→ 3, R→ 2, S→ 1
(D)P→ 5, Q→ 4, R→ 2, S→ 1

Q4·PhysicsNumericalJEE Advanced 2026

Passage: A uniform circular disk of radius 0.2 m and mass 1 kg is pivoted at its top point

C

such that it can rotate freely around

C

in the

XY

plane, as shown in the figure. Initially, when the disk is at rest, a particle of mass 20 g, travelling along negative

x

direction in the

XY

plane with speed 100 ms

^{-1}

, hits the circumference of the disk at a point

P

. After collision the particle moves along negative

y

direction at a speed of 90 ms

^{-1}

. [Given: the acceleration due to gravity

(g) = -10\hat{j}

ms

^{-2}

] Question: Amount of energy loss (in J) in the collision is:

Q5·PhysicsSingle correctJEE Advanced 2026

A solid cylinder of radius

R

rolls without slipping with a center of mass speed

v_0 = \sqrt{\frac{gR}{3}}

on a horizontal surface with a vertical edge, as shown in the figure. Here,

g

is the acceleration due to the gravity. At the moment when the cylinder loses contact with the surface due to rotation around the corner, the speed of its center of mass is:

(A)0
(B) $\sqrt{\frac{5gR}{7}}$
(C) $\sqrt{\frac{gR}{15}}$
(D) $\sqrt{\frac{3gR}{7}}$

Q6·PhysicsSingle correctJEE Main 2026

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_0 = 49

m/s, perpendicular to its axis and a rotational velocity

v_0/4R

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

m/s

^2

)

(A)15
(B)5
(C)10
(D)7.5

Q7·PhysicsSingle correctJEE Main 2026

The position of center of mass of three masses 2 kg, 3 kg and 15 kg placed with respect to mid point (p) of normal bisector, as shown in the figure is ________.

(A) $\left(\frac{\sqrt{3}}{4}, 1.25\right)$
(B) $\left(\frac{\sqrt{3}}{4}, 1.0\right)$
(C)(0, 0)
(D)(1.25, 0)

Q8·PhysicsSingle correctJEE Main 2026

Two identical bodies

A

and

B

of equal masses have initial velocities

\vec{v}_{1} = 4\hat{i}

m/s and

\vec{v}_{2} = 4\hat{j}

m/s respectively. The body

A

has acceleration

\vec{a}_{1} = 6\hat{i} + 6\hat{j}

m/s

^{2}

while the acceleration of the other body

B

is zero. The centre of mass of the two bodies moves in __________ path.

(A)circular
(B)parabolic
(C)straight line
(D)elliptical

Q9·PhysicsSingle correctJEE Main 2026

A solid sphere

(A)

of mass

5m

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

Q10·PhysicsSingle correctJEE Main 2026

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)0.128π Nm, 100
(B)0.0128π Nm, 50
(C)0.128π Nm, 50
(D)0.0128π Nm, 100

Q11·PhysicsSingle correctJEE Main 2026

An object of uniform density rolls up the curved path with the initial velocity v₀ as shown in the figure. If the maximum height attained by an object is

\dfrac{7v_{0}^{2}}{10g}

(g = acceleration due to gravity), the object is a ________.

(A)solid cylinder
(B)ring
(C)disc
(D)solid sphere

Q12·PhysicsSingle correctJEE Main 2026

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 θ₁ and in the next 2 s it rotates through an angle θ₂. The ratio

\dfrac{\theta_{2}}{\theta_{1}}

is ________.

(A)6
(B)3
(C)4
(D) $\dfrac{1}{3}$

Q13·PhysicsSingle correctJEE Main 2026

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

2R

. 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

Q14·PhysicsSingle correctJEE Main 2026

A particle is rotating in a circular path and at any instant its motion can be described as

\theta = \frac{5t^{4}}{40} - \frac{t^{3}}{3}

. The angular acceleration of the particle after 10 seconds is ______ rad/s

^{2}

.

(A)150
(B)120
(C)130
(D)170

Q15·PhysicsNumericalJEE Main 2026

Moment of inertia about an axis

AB

for a rod of mass 40 kg and length 3 m is same as that of a solid sphere of mass of 10 kg and radius

R

about an axis parallel to

AB

axis with separation of 3 m as shown in figure below. The value of

R

is given as

\sqrt{\frac{\alpha}{2}}

. The value of

\alpha

is ________.

Q16·PhysicsSingle correctJEE Main 2026

Two blocks of masses 2 kg and 1 kg respectively, are tied to the ends of a string which passes over a light frictionless pulley as shown in the figure below. The masses are held at rest at the same horizontal level and then released. The distance traversed by the centre of mass in 2 s is ______ m. (Take

g = 10

m/s

^{2}

)

(A)3.33
(B)3.12
(C)2.22
(D)1.42

Q17·PhysicsSingle correctJEE Main 2026

The position of an object having mass 0.1 kg as a function of time

t

is given as

\vec{r} = \left(10t^{2}\hat{i} + 5t^{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

Q18·PhysicsSingle correctJEE Main 2026

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)Linear momentum
(B)Velocity
(C)Acceleration
(D)Angular momentum

Q19·PhysicsNumericalJEE Main 2026

A fly wheel having mass 3 kg and radius 5 m is free to rotate about a horizontal axis. A string having negligible mass is wound around the wheel and the loose end of the string is connected to a 3 kg mass. The mass is kept at rest initially and released. Kinetic energy of the wheel when the mass descends by 3 m is _______ J. (g = 10 m/s²)

Q20·PhysicsSingle correctJEE Main 2026

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

AB

is _______ rad/s

^{2}

.

(A)22
(B)11
(C)100
(D)27

Q21·PhysicsNumericalJEE Main 2026

A solid sphere of radius 10 cm is rotating about an axis which is at a distance 15 cm from its centre. The radius of gyration about this axis is

\sqrt{n}

cm. The value of

n

is

Q22·PhysicsSingle correctJEE Main 2026

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 ________ kg.m

^{2}

.

(g = 9.8

m/s

^{2})

(A) $9.5 \times 10^{-3}$
(B) $4.75 \times 10^{-3}$
(C) $1.86 \times 10^{-2}$
(D) $8.3 \times 10^{-3}$

Q23·PhysicsSingle correctJEE Main 2026

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________. (g = gravitational acceleration)

(A) $\sqrt{\frac{3}{2}\frac{g}{L}}$
(B) $\frac{3}{\sqrt{2}}\sqrt{\frac{g}{L}}$
(C) $\frac{1}{\sqrt{2}}\sqrt{\frac{g}{L}}$
(D) $\sqrt{\frac{3g}{L}}$

Q24·PhysicsNumericalJEE Main 2026

A uniform solid cylinder of length

L

and radius

R

has moment of inertia about its axis equal to

I_1

. A small co-centric cylinder of length

L/2

and radius R/3 carved from this cylinder has moment of inertia about its axis equals to

I_2

. The ratio

I_1/I_2

is _____.

Q25·PhysicsNumericalJEE Main 2026

Suppose there is a uniform circular disc of mass M kg and radius r m shown in figure. The shaded regions are cut out from the disc. The moment of inertia of the remainder about the axis A of the disc is given by

\frac{x}{256}

M

r^{2}

. The value of x is __________.

Q26·PhysicsSingle correctJEE Main 2026

Two small balls with masses m and 2m 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) $\frac{3}{2}\frac{L}{md^2}$
(B) $\frac{2L}{md^2}$
(C) $\frac{4}{3}\frac{L}{md^2}$
(D) $\frac{2L}{5md^2}$

Q27·PhysicsSingle correctJEE Main 2026

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

(A) $\frac{3}{8}MR^2 + \frac{7}{12}ML^2$
(B) $\frac{3}{4}MR^2 + \frac{1}{6}ML^2$
(C) $\frac{3}{4}MR^2 + \frac{7}{12}ML^2$
(D) $\frac{3}{8}MR^2 + \frac{1}{6}ML^2$

Q28·PhysicsSingle correctJEE Main 2026

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 ω. The ratio of v and ω is :

(A)33
(B) $2\sqrt{88}$
(C)66
(D)32

Q29·PhysicsSingle correctJEE Main 2026

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__________kg.m

^{2}

.

(A)0.36
(B)0.72
(C)0.18
(D)0.63

Q30·PhysicsNumericalJEE Main 2026

A circular disc has radius

R_1

and thickness

T_1

. Another circular disc made of the same material has radius

R_2

and thickness

T_2

. If the moment of inertia of both discs are same and

\frac{R_1}{R_2} = 2

then

\frac{T_1}{T_2} = \frac{1}{\alpha}

. The value of α is ____.

Q31·PhysicsSingle correctJEE Main 2026

Given below are two statements : Statement I : For a mechanical system of many particles total kinetic energy is the sum of kinetic energies of all the particles. Statement II : The total kinetic energy can be the sum of kinetic energy of the center of mass w.r.t. to the origin and the kinetic energy of all the particles w.r.t. the center of mass as the reference. In the light of the above statements, choose the correct answer from the options given below :

(A)Both Statement I and Statement II are true
(B)Statement I is true but Statement II is false
(C)Statement I is false but Statement II is true
(D)Both Statement I and Statement II are false

Q32·PhysicsNumericalJEE Main 2026

Two masses m and 2m are connected by a light string going over a pulley (disc) of mass 30m with radius

r = 0.1

m. The pulley is mounted in a vertical plane and it is free to rotate about its axis. The 2m mass is released from rest and its speed when it has descended through a height of 3.6 m is ________ m/s. (Assume string does not slip and g

= 10

m/s

^2

)

Q33·PhysicsSingle correctJEE Main 2026

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) $mg/2$
(B) $mg/4$
(C) $mg/3$
(D) $mg$

Q34·PhysicsNumericalJEE Main 2026

Two identical thin rods of mass M kg and length L m are connected as shown in figure. Moment of inertia of the combined rod system about an axis passing through point P and perpendicular to the plane of the rods is

\frac{x}{2}

ML

^{2}

kg m

^{2}

. The value of x is ___________.

Q35·PhysicsSingle correctJEE Main 2026

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 centre. The magnitudes of the acceleration experienced by the blocks is____ (assume no slipping of string on pulley.)

(A) $\frac{(M-m)g}{\left[\left(\frac{13}{6}\right)M + m\right]}$
(B) $\frac{(M-m)g}{M+m}$
(C) $\frac{(M-m)g}{\left[\left(\frac{8}{3}\right)M + m\right]}$
(D) $\frac{(M-m)g}{2M+m}$

Q36·PhysicsSingle correctJEE Main 2026

Two cars A and B each of mass

10^3

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

(A) $3.6 \times 10^5$
(B) $10^5$
(C) $3 \times 10^5$
(D) $2 \times 10^5$

Q37·PhysicsSingle correctJEE Main 2025

A rod of linear mass density '

\lambda

' and length '

L

' is bent to form a ring of radius '

R

'. Moment of inertia of ring about any of its diameter is:

(A) $\dfrac{\lambda L^3}{16\pi^2}$
(B) $\dfrac{\lambda L^3}{12}$
(C) $\dfrac{\lambda L^3}{4\pi^2}$
(D) $\dfrac{\lambda L^3}{8\pi^2}$

Q38·PhysicsIntegerJEE Main 2025

A thin solid disk of 1 kg is rotating along its diameter axis at the speed of 1800 rpm. By applying an external torque of

25\pi

Nm for 40 s, the speed increases to 2100 rpm. The diameter of the disk is ___ m.

Q39·PhysicsIntegerJEE Main 2025

A, B and C are disc, solid sphere and spherical shell respectively with same radii and masses. These masses are placed as shown in the figure. The moment of inertia of the given system about PQ is

\dfrac{x}{15}I

, where

I

is the moment of inertia of the disc about its diameter. The value of

x

is ______.

Q40·PhysicsIntegerJEE Main 2025

Let M and R be the mass and radius of a disc. A small disc of radius

\dfrac{R}{3}

is removed from the bigger disc as shown in the figure. The moment of inertia of the remaining part of the bigger disc about an axis AB passing through the centre O and perpendicular to the plane of the disc is

\dfrac{4}{x}MR^2

. The value of x is __________.

Q41·PhysicsSingle correctJEE Main 2025

A rod of length 5L is bent right angle keeping one side length as 2L (as shown in the figure). The position of the centre of mass of the system: (Consider L = 10 cm)

(A) $2\hat{i}+3\hat{j}$
(B) $3\hat{i}+7\hat{j}$
(C) $5\hat{i}+8\hat{j}$
(D) $4\hat{i}+9\hat{j}$

Q42·PhysicsIntegerJEE Main 2025

A solid sphere with uniform density and radius R is rotating with constant angular velocity

\omega_1

about its diameter. After some time during the rotation its mass starts losing at a uniform rate, with no change in its shape. The angular velocity of the sphere when its radius becomes

R/2

is

x\omega_1

. The value of x is ______.

Q43·PhysicsIntegerJEE Main 2025

A circular ring and a solid sphere having same radius roll down on an inclined plane from rest without slipping. The ratio of their velocities when reached at the bottom of the plane is

\sqrt{\dfrac{x}{5}}

where

x=

______.

Q44·PhysicsSingle correctJEE Main 2025

Which of the following are correct expression for torque acting on a body? A.

\vec{\tau}=\vec{r}\times\vec{L}

B.

\vec{\tau}=\dfrac{d}{dt}(\vec{r}\times\vec{p})

C.

\vec{\tau}=\vec{r}\times\dfrac{d\vec{p}}{dt}

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)B, D and E Only
(B)C and D Only
(C)B, C, D and E Only
(D)A, B, D and E Only

Q45·PhysicsSingle correctJEE Main 2025

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

(A) $4\sqrt2$ m/s
(B)8 m/s
(C)4 m/s
(D) $8\sqrt2$ m/s

Q46·PhysicsSingle correctJEE Main 2025

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, the acceleration of the center of the sphere is:

(A) $3.5\,\text{m/s}^2$
(B) $0.35\,\text{m/s}^2$
(C) $2.5\,\text{m/s}^2$
(D) $0.25\,\text{m/s}^2$

Q47·PhysicsSingle correctJEE Main 2025

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

Q48·PhysicsSingle correctJEE Main 2025

A square Lamina OABC of length 10 cm is pivoted at ‘O’. Forces act at the lamina as shown in figure. If the lamina remains stationary, then the magnitude of

F

is:

(A)20 N
(B)0 (zero)
(C)10 N
(D) $10\sqrt{2}$ N

Q49·PhysicsIntegerJEE Main 2025

A wheel of radius 0.2 m rotates freely about its center when a string that is wrapped over its rim is pulled by a force of 10 N as shown in the figure. The established torque produces an angular acceleration of 2

rad/s^2

. Moment of inertia of the wheel is ______

kg\,m^2

. (Acceleration due to gravity

=10\,m/s^2

)

Q50·PhysicsSingle correctJEE Main 2025

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) $\dfrac{1}{2}Mr^2$
(B) $\dfrac{3}{8}Mr^2$
(C) $\dfrac{3}{2}Mr^2$
(D) $2Mr^2$

Q51·PhysicsSingle correctJEE Main 2025

Moment of inertia of a rod of mass ‘M’ and length ‘L’ about an axis passing through its centre 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 this cross about an axis passing through its centre and normal to the plane containing the cross is:

(A) $\alpha$
(B) $\dfrac{\alpha}{4}$
(C) $\dfrac{\alpha}{8}$
(D) $\dfrac{\alpha}{2}$

Q52·PhysicsSingle correctJEE Main 2025

Three equal masses m are kept at vertices (A, B, C) of an equilateral triangle of side a in free space. At t = 0, they are given an initial velocity

\vec{V}_A=V_0\,\hat{AC}

,

\vec{V}_B=V_0\,\hat{BA}

and

\vec{V}_C=V_0\,\hat{CB}

. Here,

\hat{AC}, \hat{CB}

and

\hat{BA}

are unit vectors along the edges of the triangle. If the three masses interact gravitationally, then the magnitude of the net angular momentum of the system at the point of collision is:

(A) $\dfrac{1}{2}a m V_0$
(B) $3 a m V_0$
(C) $\dfrac{\sqrt{3}}{2}a m V_0$
(D) $\dfrac{3}{2}a m V_0$

Q53·PhysicsIntegerJEE Main 2025

The coordinates of a particle with respect to origin in a given reference frame is (1, 1, 1) meters. If a force

\vec{F}=\hat{i}-\hat{j}+\hat{k}

acts on the particle, then the magnitude of torque (with respect to origin) in z-direction is ______ Nm.

Q54·PhysicsSingle correctJEE Main 2025

The centre of mass of a thin rectangular plate with sides of length a and b, whose mass per unit area

(\sigma)

varies as

\sigma=\frac{\sigma_0 x}{ab}

(where

\sigma_0

is a constant), would be______

(A) $\left(\frac{2}{3}a, \frac{b}{2}\right)$
(B) $\left(\frac{2}{3}a, \frac{2}{3}b\right)$
(C) $\left(\frac{a}{2}, \frac{b}{2}\right)$
(D) $\left(\frac{1}{3}a, \frac{b}{2}\right)$

Q55·PhysicsIntegerJEE Main 2025

The moment of inertia of a solid disc rotating along its diameter is 2.5 times higher than the moment of inertia of a ring rotating in similar way. The moment of inertia of a solid sphere which has same radius as the disc and rotating in similar way, is n times higher than the moment of inertia of the given ring. Here, n = _______. Consider all the bodies have equal masses.

Q56·PhysicsSingle correctJEE Main 2025

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)300 g
(B)190 g
(C)200 g
(D)290 g

Q57·PhysicsIntegerJEE Main 2025

Two iron solid discs of negligible thickness have radii

R_1

and

R_2

and moment of inertia

I_1

and

I_2

, respectively. For

R_2=2R_1

, the ratio of

I_1

and

I_2

would be

\frac{1}{x}

, where x = _______.

Q58·PhysicsSingle correctJEE Main 2025

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) $\dfrac{2}{5}$
(B) $\dfrac{5}{2}$
(C) $\dfrac{3}{4}$
(D) $\dfrac{4}{3}$

Q59·PhysicsSingle correctJEE Main 2025

An object of mass ‘m’ is projected from origin in a vertical x-y plane at an angle 45° with the x-axis with an initial velocity

v_0

. The magnitude and direction of the angular momentum of the object with respect to origin, when it reaches at the maximum height, is [g is acceleration due to gravity]

(A) $\dfrac{mv_0^3}{2\sqrt2\,g}$ along negative z-axis
(B) $\dfrac{mv_0^3}{2\sqrt2\,g}$ along positive z-axis
(C) $\dfrac{mv_0^3}{4\sqrt2\,g}$ along positive z-axis
(D) $\dfrac{mv_0^3}{4\sqrt2\,g}$ along negative z-axis

Q60·PhysicsSingle correctJEE Main 2025

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) $t_1=t_2$
(C) $t_1=2t_2$
(D) $t_1>t_2$

Q61·PhysicsSingle correctJEE Main 2025

A uniform solid cylinder of mass ‘m’ and radius ‘r’ rolls along an inclined rough plane of inclination 45°. If it starts to roll from rest from the top of the plane then the linear acceleration of the cylinder axis will be :-

(A) $\dfrac{1}{\sqrt2}g$
(B) $\dfrac{1}{3\sqrt2}g$
(C) $\dfrac{\sqrt2 g}{3}$
(D) $\sqrt2 g$

Q62·PhysicsSingle correctJEE Main 2025

A circular disk of radius R meter and mass M kg is rotating about the axis perpendicular to the disk. An external torque is applied to the disk such that

\theta(t)=5t^2-8t

, 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

s?

(A) $60\,MR^2$
(B) $72\,MR^2$
(C) $108\,MR^2$
(D) $8\,MR^2$

Q63·PhysicsSingle correctJEE Main 2025

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° 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° while keeping L constant. Then the new speed of the sphere at the bottom of the plane is

v_2

. The ratio of

v_1^2:v_2^2

is

(A) $1:\sqrt2$
(B) $1:3$
(C) $1:2$
(D) $1:\sqrt3$

Q64·PhysicsSingle correctJEE Main 2025

Consider a circular disc of radius 20 cm with centre located at the origin. A circular hole of radius 5 cm is cut from this disc in such a way that the edge of the hole touches the edge of the disc. The distance of centre of mass of residual or remaining disc from the origin will be-

(A)2.0 cm
(B)0.5 cm
(C)1.5 cm
(D)1.0 cm

Q65·PhysicsSingle correctJEE Main 2025

The torque due to the force

\left(2\hat{i}+\hat{j}+2\hat{k}\right)

about the origin, acting on a particle whose position vector is

\left(\hat{i}+\hat{j}+\hat{k}\right)

, would be

(A) $\hat{i}-\hat{j}+\hat{k}$
(B) $\hat{i}+\hat{k}$
(C) $\hat{i}-\hat{k}$
(D) $\hat{j}-\hat{k}$

Q66·PhysicsIntegerJEE Main 2025

The position vectors of two 1 kg particles A and B are given by r_A = (α₁t i + α₂t j + α₃t k) m and r_B = (β₁ i + β₂t² j + β₃t k) m, where (α₁ = 1, α₂ = 3 m/s, α₃ = 2 m/s) and (β₁ = 1 m, β₂ = ... , β₃ = ... ) are constants, with t in seconds. At t = 1 s the magnitude of the angular momentum of particle B with respect to the position of particle A is √L kg m²/s. The value of L is ______.

Q67·PhysicsSingle correctJEE Main 2025

A uniform circular disc of radius R and mass M 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 (its boundary passes through the centre). The moment of inertia of the remaining part about the same axis is:

(A)7/32 MR²
(B)9/32 MR²
(C)17/32 MR²
(D)13/32 MR²

Q68·PhysicsIntegerJEE Main 2025

A tube of length 1m is filled completely with an ideal liquid of mass 2M, and closed at both ends. The tube is rotated uniformly in horizontal plane about one of its ends. If the force exerted by the liquid at the other end is F then angular velocity of the tube is

\sqrt{\frac{F}{\alpha M}}

in SI unit. The value of

\alpha

is ______.

Q69·PhysicsIntegerJEE Advanced 2024

A disc of mass

M

and radius

R

is free to rotate about its vertical axis as shown in the figure. A battery operated motor of negligible mass is fixed to this disc at a point on its circumference. Another disc of the same mass

M

and radius

R/2

is fixed to the motor's thin shaft. Initially, both the discs are at rest. The motor is switched on so that the smaller disc rotates at a uniform angular speed

\omega

. If the angular speed at which the large disc rotates is

\omega/n

, then the value of

n

is ______.

Q70·PhysicsIntegerJEE Advanced 2024

A thin uniform rod of length

L

and certain mass is kept on a frictionless horizontal table with a massless string of length

L

fixed to one end (top view is shown in the figure). The other end of the string is pivoted to a point

O

. If a horizontal impulse

P

is imparted to the rod at a distance

x = L/n

from the mid-point of the rod (see figure), then the rod and string revolve together around the point

O

, with the rod remaining aligned with the string. In such a case, the value of

n

is ______.

Q71·PhysicsSingle correctJEE Main 2024

A heavy iron bar, of weight

W

is having its one end on the ground and the other on the shoulder of a person. The bar makes an angle

\theta

with the horizontal. The weight experienced by the person is:

(A) $\frac{W}{2}$
(B) $W$
(C) $W\cos\theta$
(D) $W\sin\theta$

Q72·PhysicsNumericalJEE Main 2024

A string is wrapped around the rim of a wheel of moment of inertia

0.40\ \mathrm{kgm^{2}}

and radius 10 cm. The wheel is free to rotate about its axis. Initially the wheel is at rest. The string is now pulled by a force of 40 N. The angular velocity of the wheel after 10 s is

x

rad/s, where

x

is ______.

Q73·PhysicsNumericalJEE Main 2024

A circular disc reaches from top to bottom of an inclined plane of length

l

. When it slips down the plane, it takes time

t_{1}

. When it rolls down the plane then it takes

\left(\dfrac{\alpha}{2}\right)^{1/2}t_{1}

s, where

\alpha

is _______.

Q74·PhysicsSingle correctJEE Main 2024

A thin circular disc of mass M and radius 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

\dfrac{M}{2}

is placed gently on the first disc co-axially, then the new angular velocity of the system is:

(A) $\dfrac{4\omega}{5}$
(B) $\dfrac{5\omega}{4}$
(C) $\dfrac{2\omega}{3}$
(D) $\dfrac{3\omega}{2}$

Q75·PhysicsNumericalJEE Main 2024

A uniform thin metal plate of mass

10\,kg

with dimensions as shown. The ratio of

x

and

y

coordinates of center of mass of plate is

\dfrac{n}{9}

. The value of

n

is ___

Q76·PhysicsNumericalJEE Main 2024

A circular table is rotating with an angular velocity of

\omega

rad/s about its axis. There is a smooth groove along a radial direction on the table. A steel ball is gently placed at a distance of 1 m on the groove. All the surfaces are smooth. If the radius of the table is 3 m, the radial velocity of the ball w.r.t. the table at the time the ball leaves the table is

x\sqrt{2}

m/s, where the value of x is ________ .

Q77·PhysicsNumericalJEE Main 2024

Three balls of masses 2kg, 4kg and 6kg respectively are arranged at centre of the edges of an equilateral triangle of side 2 m. The moment of inertia of the system about an axis through the centroid and perpendicular to the plane of triangle, will be ______ kg m

^{2}

.

Q78·PhysicsNumericalJEE Main 2024

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

Q79·PhysicsSingle correctJEE Main 2024

The ratio of the radius of gyration of a hollow sphere to that of a solid cylinder of equal mass, for the moment of inertia about their diameter axis AB as shown in the figure, is

\sqrt{\dfrac{8}{x}}

. The value of

x

is:

(A)34
(B)17
(C)67
(D)51

Q80·PhysicsNumericalJEE Main 2024

A hollow sphere is rolling on a plane surface about its axis of symmetry. The ratio of its rotational kinetic energy to its total kinetic energy is

\dfrac{x}{5}

. The value of

x

is __________.

Q81·PhysicsNumericalJEE Main 2024

In a system two particles of masses

m_1=3

kg and

m_2=2

kg are placed at certain distance from each other. The particle of mass

m_1

is moved towards the center of mass of the system through a distance 2 cm. In order to keep the center of mass of the system at the original position, the particle of mass

m_2

should move towards the center of mass by the distance ______ cm.

Q82·PhysicsNumericalJEE Main 2024

A solid sphere and a hollow cylinder roll up without slipping on same inclined plane with same initial speed

v

. The sphere and the cylinder reaches upto maximum heights

h_1

and

h_2

, respectively, above the initial level. The ratio

h_1:h_2

is

\dfrac{n}{10}

. The value of

n

is ___

Q83·PhysicsSingle correctJEE Main 2024

A disc of radius R and mass 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) $\dfrac{v^2}{g}$
(B) $\dfrac{3}{4}\dfrac{v^2}{g}$
(C) $\dfrac{1}{2}\dfrac{v^2}{g}$
(D) $\dfrac{2}{3}\dfrac{v^2}{g}$

Q84·PhysicsNumericalJEE Main 2024

A uniform rod AB of mass 2 kg and Length 30 cm at rest on a smooth horizontal surface. An impulse of force 0.2 Ns is applied to end B. The time taken by the rod to turn through at right angles will be

\dfrac{\pi}{x}

s, where

x=

__________.

Q85·PhysicsNumericalJEE Main 2024

The identical spheres each of mass

2M

are placed at the corners of a right angled triangle with mutually perpendicular sides equal to

4\,m

each. Taking point of intersection of these two sides as origin, the magnitude of position vector of the centre of mass of the system is

\dfrac{4\sqrt2}{x}

, where the value of

x

is ___

Q86·PhysicsNumericalJEE Main 2024

A body of mass '

m

' is projected with a speed '

u

' making an angle of

45^\circ

with the ground. The angular momentum of the body about the point of projection, at the highest point is expressed as

\dfrac{\sqrt{2}\,mu^3}{Xg}

. The value of '

X

' is ______.

Q87·PhysicsNumericalJEE Main 2024

Two identical spheres each of mass 2 kg and radius 50 cm are fixed at the ends of a light rod so that the separation between the centers is 150 cm. Then, moment of inertia of the system about an axis perpendicular to the rod and passing through its middle point is

\dfrac{x}{20}

kg m

^2

, where the value of

x

is ______.

Q88·PhysicsNumericalJEE Main 2024

A solid circular disc of mass 50 kg rolls along a horizontal floor so that its center of mass has a speed of 0.4 m/s. The absolute value of work done on the disc to stop it is ______ J.

Q89·PhysicsSingle correctJEE Main 2024

A particle of mass

m

is projected with a velocity

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

h

is:

(A) $\dfrac{\sqrt3\,mu^3}{16g}$
(B) $\dfrac{\sqrt3\,mu^3}{2g}$
(C) $\dfrac{mu^3}{\sqrt2\,g}$
(D)zero

Q90·PhysicsNumericalJEE Main 2024

Two discs of moment of inertia

I_1=4

kg m

^2

and

I_2=2

kg m

^2

about their central axes & normal to their planes, rotating with angular speeds 10 rad/s & 4 rad/s respectively are brought into contact face to face with their axes of rotation coincident. The loss in kinetic energy of the system in the process is ______ J.

Q91·PhysicsNumericalJEE Main 2024

Consider a Disc of mass

5\,kg

, radius

2\,m

, rotating with angular velocity of

10\,rad/s

about an axis perpendicular to the plane of rotation. An identical disc is kept gently over the rotating disc along the same axis. The energy dissipated so that the discs continue to rotate together without slipping is ___

J

.

Q92·PhysicsNumericalJEE Main 2024

A cylinder is rolling down on an inclined plane of inclination

60^\circ

. It’s acceleration during rolling down will be

\dfrac{x}{\sqrt{3}}

m/s

^2

, where

x=

______ . (use

g=10

m/s

^2

).

Q93·PhysicsNumericalJEE Main 2024

A body of mass 5 kg is moving with a uniform speed

3\sqrt{2}

ms

^{-1}

in X–Y plane along the line

y=x+4

. The angular momentum of the particle about the origin will be ___ kg m

^{2}

s

^{-1}

.

Q94·PhysicsNumericalJEE Main 2024

A ring and a solid sphere roll down the same inclined plane without slipping. They start from rest. The radii of both bodies are identical and the ratio of their kinetic energies is

\dfrac{7}{x}

where x is ________.

Q95·PhysicsNumericalJEE Main 2024

Four particles each of mass 1 kg are placed at four corners of a square of side 2 m. Moment of inertia of system about an axis perpendicular to its plane and passing through one of its vertex is __________ kgm

^2

.

Q96·PhysicsIntegerJEE Advanced 2023

A thin circular coin of mass 5 gm and radius 4/3 cm is initially in a horizontal xy -plane. The coin is tossed vertically up (+z direction) by applying an impulse of

\sqrt{\frac{\pi}{2}}\times10^{-2}

N-s at a distance 2/3 cm from its center. The coin spins about its diameter and moves along the +z direction. By the time the coin reaches back to its initial position, it completes n rotations. The value of n is ________. [Given: The acceleration due to gravity g = 10 m s

^{-2}

]

Q97·PhysicsMultiple correctJEE Advanced 2023

An annular disk of mass M, inner radius a and outer radius b is placed on a horizontal surface with coefficient of friction

\mu

, as shown in the figure. At some time, an impulse

J_0\hat{x}

is applied at a height h above the center of the disk. If

h = h_m

then the disk rolls without slipping along the x-axis. Which of the following statement(s) is(are) correct?

(A)For $\mu \neq 0$ and $a \rightarrow 0$ , $h_m = b/2$
(B)For $\mu \neq 0$ and $a \rightarrow b$ , $h_m = b$
(C)For $h = h_m$ , the initial angular velocity does not depend on the inner radius a.
(D)For $\mu = 0$ and $h = 0$ , the wheel always slides without rolling.

Q98·PhysicsSingle correctJEE Advanced 2023

A bar of mass M = 1.00 kg and length L = 0.20 m is lying on a horizontal frictionless surface. One end of the bar is pivoted at a point about which it is free to rotate. A small mass m = 0.10 kg is moving on the same horizontal surface with 5.00

ms^{-1}

speed on a path perpendicular to the bar. It hits the bar at a distance L/2 from the pivoted end and returns back on the same path with speed v. After this elastic collision, the bar rotates with an angular velocity

\omega

. Which of the following statement is correct?

(A) $\omega = 6.98$ rad $s^{-1}$ and $v = 4.30\ ms^{-1}$
(B) $\omega = 3.75$ rad $s^{-1}$ and $v = 4.30\ ms^{-1}$
(C) $\omega = 3.75$ rad $s^{-1}$ and $v = 10.0\ ms^{-1}$
(D) $\omega = 6.80$ rad $s^{-1}$ and $v = 4.10\ ms^{-1}$

Q99·PhysicsNumericalJEE Main 2023

A solid sphere and a solid cylinder of same mass and radius are rolling on a horizontal surface without slipping. The ratio of their radius of gyrations respectively (

k_{sph} : k_{cyl}

) is

2 : \sqrt{x}

, then value of x is __________.

Q100·PhysicsSingle correctJEE Main 2023

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

A

as shown in figure. When the disc completes half of its rotation, the displacement of point

A

from its initial position is:

(A) $R\sqrt{\pi^{2}+4}$
(B) $R\sqrt{\pi^{2}+1}$
(C) $2R$
(D) $2R\sqrt{1+4\pi^{2}}$

Q101·PhysicsNumericalJEE Main 2023

A light rope is wound around a hollow cylinder of mass 5 kg and radius 70 cm. The rope is pulled with a force of 52.5 N. The angular acceleration of the cylinder will be ____ rad s

^{-2}

.

Q102·PhysicsNumericalJEE Main 2023

A solid sphere is rolling on a horizontal plane without slipping. If the ratio of angular momentum about axis of rotation of the sphere to the total energy of moving sphere is

\pi:22

then, the value of its angular speed will be _____ rad/s.

Q103·PhysicsNumericalJEE Main 2023

For a rolling spherical shell, the ratio of rotational kinetic energy and total kinetic energy is

\dfrac{x}{5}

. The value of x is _________.

Q104·PhysicsNumericalJEE Main 2023

A circular plate is rotating in a horizontal plane, about an axis passing through its centre and perpendicular to the plate, with an angular velocity

\omega

. A person sits at the centre having two dumbbells in his hands. When he stretches out his hands the moment of inertia of the system becomes triple. If E is the initial kinetic energy of the system, then the final kinetic energy will be

\dfrac{E}{x}

. The value of x is ____.

Q105·PhysicsNumericalJEE Main 2023

A solid sphere of mass 500 g and radius 5 cm is rotated about one of its diameter with angular speed of 10 rad

\text{s}^{-1}

. If the moment of inertia of the sphere about its tangent is

x \times 10^{-2}

times its angular momentum about the diameter. Then the value of x will be ______.

Q106·PhysicsNumericalJEE Main 2023

A force of

-P\hat{k}

acts on the origin of the coordinate system. The torque about the point

(2,-3)

is

P(a\hat{i}+b\hat{j})

. If the ratio

\tfrac{a}{b}=\tfrac{x}{2}

, then the value of x is _______ .

Q107·PhysicsAssertion & ReasonJEE Main 2023

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason 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 the above statements, choose the most appropriate answer from the options given below:

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

Q108·PhysicsNumericalJEE Main 2023

If the earth suddenly shrinks to

\dfrac{1}{64}

th of its original volume with its mass remaining the same, the period of rotation of earth becomes

\dfrac{24}{x}

h. The value of x is _________.

Q109·PhysicsNumericalJEE Main 2023

The moment of inertia of semicircular ring about an axis, passing through the center and perpendicular to the plane of ring, is

\frac{1}{x}MR^{2}

, where

R

is the radius and

M

is the mass of semicircular ring. The value of

x

will be

Q110·PhysicsNumericalJEE Main 2023

A hollow spherical ball of uniform density rolls up a curved surface with an initial velocity 3 m/s (as shown in the figure). The maximum height with respect to the initial position covered by it will be ____ cm.

Q111·PhysicsNumericalJEE Main 2023

Two identical solid spheres each of mass

2\,kg

and radius

10\,cm

are fixed at the ends of a light rod. The separation between the centres of the spheres is

40\,cm

. The moment of inertia of the system about an axis perpendicular to the rod passing through its middle point is _____

\times10^{-3}\,kg\,m^{2}

.

Q112·PhysicsNumericalJEE Main 2023

A ring and a solid sphere rotating about an axis passing through their centers have same radii of gyration. The axis of rotation is perpendicular to plane of ring. The ratio of radius of ring to that of sphere is

\sqrt{\dfrac{2}{x}}

. The value of

x

is

Q113·PhysicsNumericalJEE Main 2023

Moment of inertia of a disc of mass

M

and radius '

R

' about any of its diameter is

\dfrac{MR^2}{4}

. The moment of inertia of this disc about an axis normal to the disc and passing through a point on its edge will be

\dfrac{x}{2}MR^2

. The value of

x

is _________.

Q114·PhysicsNumericalJEE Main 2023

A solid cylinder is released from rest from the top of an inclined plane of inclination

30^\circ

and length

60

cm. If the cylinder rolls without slipping, its speed upon reaching the bottom of the inclined plane is _______ ms

^{-1}

(given

g=10

ms

^{-2}

).

Q115·PhysicsNumericalJEE Main 2023

A solid sphere of mass

1

kg rolls without slipping on a plane surface. Its kinetic energy is

7 \times 10^{-3}

J. The speed of the centre of mass of the sphere is __________

\text{cms}^{-1}

Q116·PhysicsNumericalJEE Main 2023

Two discs of same mass and different radii are made of different materials such that their thicknesses are

1

cm and

0.5

cm respectively. The densities of materials are in the ratio

3:5

. The moment of inertia of these discs respectively about their diameters will be in the ratio of

\dfrac{x}{6}

. The value of

x

is _________.

Q117·PhysicsNumericalJEE Main 2023

A uniform disc of mass 0.5 kg and radius

r

is projected with velocity 18 m/s at

t=0

s on a rough horizontal surface. It starts with a purely sliding motion at

t=0

s and after 2 s it acquires a purely rolling motion. The total kinetic energy of the disc after 2 s will be _______ J. (Given coefficient of friction is 0.3 and

g=10

m/s

^2

)

Q118·PhysicsNumericalJEE Main 2023

A thin uniform rod of length

2

m, cross sectional area '

A

' and density '

d

' is rotated about an axis passing through the centre and perpendicular to its length with angular velocity

\omega

. If value of

\omega

in terms of its rotational kinetic energy

E

is

\sqrt{\dfrac{\alpha E}{Ad}}

then value of

\alpha

is

Q119·PhysicsNumericalJEE Main 2023

A solid sphere of mass 2 kg is making pure rolling on a horizontal surface with kinetic energy 2240 J. The velocity of centre of mass of the sphere will be ________ ms

^{-1}

.

Q120·PhysicsNumericalJEE Main 2023

A particle of mass

100\,g

is projected at time

t=0

with a speed

20\,ms^{-1}

at an angle

45^\circ

to the horizontal as given in the figure. The magnitude of the angular momentum of the particle about the starting point at time

t=2\,s

is found to be

\sqrt K\,kg\,m^2/s

. The value of

K

is _____. (Take

g=10\,ms^{-2}

)

Q121·PhysicsSingle correctJEE Main 2023

An object of mass 8 kg 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, is supported by a cable AB as shown in the figure such that the system is in equilibrium. The tension in the cable is: (Take

g=10

m/s

^2

)

(A) $90$ N
(B) $30$ N
(C) $300$ N
(D) $240$ N

Q122·PhysicsNumericalJEE Main 2023

If a solid sphere of mass 5 kg and a disc of mass 4 kg have the same radius. Then the ratio of moment of inertia of the disc about a tangent in its plane to the moment of inertia of the sphere about its tangent will be

\frac{x}{7}

. The value of

x

is ______.

Q123·PhysicsNumericalJEE Main 2023

I_{CM}

is the moment of inertia of a circular disc about an axis (CM) passing through its center and perpendicular to the plane of the disc.

I_{AB}

is its moment of inertia about an axis AB perpendicular to the plane and parallel to axis CM at a distance

\dfrac{2}{3}R

from the center, where R is the radius of the disc. The ratio of

I_{AB}

and

I_{CM}

is

x:9

. The value of

x

is _______.

Q124·PhysicsNumericalJEE Main 2023

A uniform solid cylinder with radius

R

and length

L

has moment of inertia

I_1

about the axis of the cylinder. A concentric solid cylinder of radius

R'=\dfrac{R}{2}

and length

L'=\dfrac{L}{2}

is carved out of the original cylinder. If

I_2

is the moment of inertia of the carved out portion of the cylinder then

\dfrac{I_1}{I_2}=

_________ (Both

I_1

and

I_2

are about the axis of the cylinder)

Q125·PhysicsNumericalJEE Main 2023

Solid sphere A is rotating about an axis PQ. If the radius of the sphere is 5 cm then its radius of gyration about PQ will be

\sqrt{x}

cm. The value of

x

is _______ .

Q126·PhysicsNumericalJEE Advanced 2022

A solid sphere of mass 1 kg and radius 1 m rolls without slipping on a fixed inclined plane with an angle of inclination

\theta=30^\circ

from the horizontal. Two forces of magnitude 1N each, parallel to the incline, act on the sphere, both at distance r = 0.5 m from the center of the sphere, as shown in the figure. The acceleration of the sphere down the plane is ________ ms

^{-2}

. (Take g = 10 ms

^{-2}

.)

Q127·PhysicsIntegerJEE Advanced 2022

A particle of mass 1 kg is subjected to a force which depends on the position as

\vec{F} = -k(x\hat{i} + y\hat{j})

kg ms

^{-2}

with k = 1 kg s

^{-2}

. At time t = 0, the particle's position

\vec{r} = \left(\frac{1}{\sqrt{2}}\hat{i} + \sqrt{2}\hat{j}\right)

m and its velocity

\vec{v} = \left(-\sqrt{2}\hat{i} + \sqrt{2}\hat{j} + \frac{2}{\pi}\hat{k}\right)

ms

^{-1}

, Let v

_x

and v

_y

denote the x and the y components of the particle's velocity, respectively. Ignore gravity. When z = 0.5 m, the value of (x v

_y

− y v

_x

) is ______ m

^{2}

s

^{-1}

.

Q128·PhysicsSingle correctJEE Advanced 2022

A flat surface of a thin uniform disk A of radius R is glued to a horizontal table. Another thin uniform disk B of mass M and with the same radius R rolls without slipping on the circumference of A , as shown in the figure. A flat surface of B also lies on the plane of the table. The center of mass of B has fixed angular speed

\omega

about the vertical axis passing through the center of A . The angular momentum of B is

nM\omega R^2

with respect to the center of A . Which of the following is the value of n ?

(A)2
(B)5
(C)7/2
(D)9/2

Q129·PhysicsNumericalJEE Advanced 2022

At time t=0, a disk of radius 1 m starts to roll without slipping on a horizontal plane with an angular acceleration of

\alpha = \frac{2}{3}\ rad\ s^{-2}

. A small stone is stuck to the disk. At t = 0, it is at the contact point of the disk and the plane. Later, at time

t = \sqrt{\pi}

s, the stone detaches itself and flies off tangentially from the disk. The maximum height (in m) reached by the stone measured from the plane is

\frac{1}{2} + \frac{x}{10}

. The value of x is______ [Take g= 10ms

^{-2}

.]

Q130·PhysicsSingle correctJEE Main 2022

The torque of a force

5\hat{i} + 3\hat{j} - 7\hat{k}

about the origin is τ. If the force acts on a particle whose position vector is

2\hat{i} + 2\hat{j} + \hat{k}

, then the value of τ 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) $17\hat{i} + 9\hat{j} + 16\hat{k}$

Q131·PhysicsSingle correctJEE Main 2022

Two bodies of mass 1 kg and 3 kg have position vectors

\hat{i} + 2\hat{j} + \hat{k}

and

-3\hat{i} - 2\hat{j} + \hat{k}

respectively. The magnitude of position vector of centre of mass of this system will be similar to the magnitude of vector :

(A) $\hat{i} - 2\hat{j} + \hat{k}$
(B) $-3\hat{i} - 2\hat{j} + \hat{k}$
(C) $-2\hat{i} + 2\hat{k}$
(D) $-2\hat{i} - \hat{j} + 2\hat{k}$

Q132·PhysicsNumericalJEE Main 2022

Four identical discs each of mass 'M' and diameter 'a' are arranged in a small plane as shown in figure. If the moment of inertia of the system about OO' is

\frac{x}{4}

Ma

^{2}

. Then, the value of x will be ________.

Q133·PhysicsNumericalJEE Main 2022

The distance of centre of mass from end A of a one dimensional rod (AB) having mass density

\rho = \rho_0\left(1 - \dfrac{x^2}{L^2}\right)

kg/m and length L (in meter) is

\dfrac{3L}{\alpha}

m. The value of

\alpha

is ......... (where x is the distance form end A)

Q134·PhysicsNumericalJEE Main 2022

A pulley of radius 1.5 m is rotated about its axis by a force

F = (12t - 3t^{2})

N applied tangentially (while t is measured in seconds). If moment of inertia of the pulley about its axis of rotation is 4.5 kg

\mathrm{m}^{2}

, the number of rotations made by the pulley before its direction of motion is reversed, will be

\frac{K}{\pi}

. The value of K is ____ .

Q135·PhysicsNumericalJEE Main 2022

A solid cylinder length is suspended symmetrically through two massless strings, as shown in the figure. The distance from the initial rest position, the cylinder should by unbinding the strings to achieve a speed of 4 ms

^{-1}

, is…….cm. (take g = 10 ms

^{-2}

)

Q136·PhysicsNumericalJEE Main 2022

The radius of gyration of a cylindrical rod about an axis of rotation perpendicular to its length and passing through the center will be________ m. Given, the length of the rod is

10\sqrt{3}

m.

Q137·PhysicsNumericalJEE Main 2022

A disc of mass 1 kg and radius R is free of rotate about a horizontal axis passing through its centre and perpendicular to the plane of disc. A body of same mass as that of disc is fixed at the highest point of the disc. Now the system is released, when the body comes to the lowest position, its angular speed will be

4\sqrt{\frac{x}{3R}}

rad s

^{-1}

where x = ______ (g = 10ms

^{-2}

)

Q138·PhysicsSingle correctJEE Main 2022

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

Q139·PhysicsSingle correctJEE Main 2022

A spherical shell of 1 kg mass and radius R is rolling with angular speed ω on horizontal plane (as shown in figure). The magnitude of angular momentum of the shell about the origin O is

\frac{a}{3}R^2\omega

. The value of a will be :

(A)2
(B)3
(C)5
(D)4

Q140·PhysicsNumericalJEE Main 2022

The moment of inertia of a uniform thin rod about a perpendicular axis passing through one end is

I_1

. The same rod is bent into a ring and its moment of inertia about a diameter is

I_2

. If

\frac{I_1}{I_2}

is

\frac{x\pi^2}{3}

, then the value of x will be________.

Q141·PhysicsSingle correctJEE Main 2022

A ball is spun with angular acceleration

\alpha = 6t^2 - 2t

where t is in second and α 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) $\frac{3}{2} t^4 - t^2 + 10t$
(B) $\frac{t^4}{2} - \frac{t^3}{3} + 10t + 4$
(C) $\frac{2t^4}{3} - \frac{t^3}{6} + 10t + 12$
(D) $2t^4 - \frac{t^3}{2} + 5t + 4$

Q142·PhysicsNumericalJEE Main 2022

The position vector of 1 kg object is

\vec{r} = \left(3\hat{i} - \hat{j}\right)

m and its velocity

\vec{v} = \left(3\hat{j} + k\right)

ms

^{-1}

. The magnitude of its angular momentum is

\sqrt{x}

Nm where x is ______ .

Q143·PhysicsNumericalJEE Main 2022

A uniform disc with mass M = 4 kg and radius R = 10 cm is mounted on a fixed horizontal axle as shown in figure. A block with mass m = 2 kg hangs from a massless cord that is wrapped around the rim of the disc. During the fall of the block, the cord does not slip and there is no friction at the axle. The tension in the cord is ________N. (Take g = 10 ms

^{-2}

)

Q144·PhysicsSingle correctJEE Main 2022

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 F

_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/s

^2

)

(A) $\frac{6}{\sqrt{110}}$
(B) $\frac{3}{\sqrt{113}}$
(C) $\frac{3}{\sqrt{109}}$
(D) $\frac{2}{\sqrt{109}}$

Q145·PhysicsSingle correctJEE Main 2022

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 :

(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

Q146·PhysicsNumericalJEE Main 2022

A rolling wheel of 12 kg is on an inclined plane at position P and connected to a mass of 3 kg through a string of fixed length and pulley as shown in figure. Consider PR as friction free surface. The velocity of centre of mass of the wheel when it reaches at the bottom Q of the inclined plane PQ will be

\dfrac{1}{2}\sqrt{xgh}

m/s. The value of x is _______ .

Q147·PhysicsSingle correctJEE Main 2022

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) $\frac{M}{\left(M+m\right)}$
(B) $\frac{\left(M+2m\right)}{2M}$
(C) $\frac{2M}{\left(M+2m\right)}$
(D) $\frac{2\left(M+2m\right)}{M}$

Q148·PhysicsSingle correctJEE Main 2022

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) $\frac{2}{5}$
(B) $\frac{2}{7}$
(C) $\frac{1}{5}$
(D) $\frac{7}{10}$

Q149·PhysicsNumericalJEE Main 2022

Moment of Inertia (M.I.) of four bodies having same mass 'M' and radius '2R' are as follows:

I_1

= M.I. of solid sphere about its diameter

I_2

= M.I. of solid cylinder about its axis

I_3

= M.I. of solid circular disc about its diameter

I_4

= M.I. of thin circular ring about its diameter If

2(I_2 + I_3) + I_4 = x. I_1

then the value of x will be ____

Q150·PhysicsSingle correctJEE Main 2022

If force

\vec{F}=3\hat{i}+4\hat{j}-2\hat{k}

acts on a particle having position vector

2\hat{i}+\hat{j}+2\hat{k}

then, the torque about the origin will be :-

(A) $3\hat{i}+4\hat{j}-2\hat{k}$
(B) $-10\hat{i}+10\hat{j}+5\hat{k}$
(C) $10\hat{i}+5\hat{j}-10\hat{k}$
(D) $10\hat{i}+\hat{j}-5\hat{k}$

Q151·PhysicsSingle correctJEE Main 2022

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

(A)7.5 rad
(B)15 rad
(C)20 rad
(D)30 rad

Q152·PhysicsNumericalJEE Main 2022

A metre scale is balanced on a knife edge at its centre. When two coins, each of mass 10 g are put one on the top of the other at the 10.0 cm mark the scale is found to be balanced at 40.0 cm mark. The mass of the metre scale is found to be

x \times 10^{-2}

kg. The value of x is

Q153·PhysicsNumericalJEE Advanced 2021

A pendulum consists of a bob of mass

m

= 0.1 kg and a massless inextensible string of length

L

= 1.0 m. It is suspended from a fixed point at height

H

= 0.9 m above a frictionless horizontal floor. Initially, the bob of the pendulum is lying on the floor at rest vertically below the point of suspension. A horizontal impulse

P

= 0.2 kg-m/s is imparted to the bob at some instant. After the bob slides for some distance, the string becomes taut and the bob lifts off the floor. The magnitude of the angular momentum of the pendulum about the point of suspension just before the bob lifts off is

J

kg-m

^2

/s. The kinetic energy of the pendulum just after the lift-off is

K

Joules. The value of

J

is ____.

Q154·PhysicsMultiple correctJEE Advanced 2021

A horizontal force

F

is applied at the center of mass of a cylindrical object of mass m and radius

R

, perpendicular to its axis as shown in the figure. The coefficient of friction between the object and the ground is

\mu

. The center of mass of the object has an acceleration

a

. The acceleration due to gravity is

g

. Given that the object rolls without slipping, which of the following statement(s) is(are) correct?

(A)For the same $F$ , the value of $a$ does not depend on whether the cylinder is solid or hollow
(B)For a solid cylinder, the maximum possible value of $a$ is $2\mu g$
(C)The magnitude of the frictional force on the object due to the ground is always $\mu mg$
(D)For a thin-walled hollow cylinder, $a = \frac{F}{2m}$

Q155·PhysicsMultiple correctJEE Advanced 2021

One end of a horizontal uniform beam of weight

W

and length

L

is hinged on a vertical wall at point

O

and its other end is supported by a light inextensible rope. The other end of the rope is fixed at point

Q

, at a height

L

above the hinge at point

O

. A block of weight

\alpha W

is attached at the point

P

of the beam, as shown in the figure (not to scale). The rope can sustain a maximum tension of

(2\sqrt{2})W

. Which of the following statement(s) is(are) correct ?

(A)The vertical component of reaction force at $O$ does not depend on $\alpha$
(B)The horizontal component of reaction force at $O$ is equal to $W$ for $\alpha = 0.5$
(C)The tension in the rope is $2W$ for $\alpha = 0.5$
(D)The rope breaks if $\alpha > 1.5$

Q156·PhysicsMultiple correctJEE Advanced 2021

A particle of mass

M

= 0.2 kg is initially at rest in the

xy

-plane at a point (x =

-l

, y =

-h

), where

l

= 10 m and

h

= 1m. The particle is accelerated at time t = 0 with a constant acceleration a = 10 m/s

^2

along the positive x-direction. Its angular momentum and torque with respect to the origin, in SI units, are represented by

\vec{L}

and

\vec{\tau}

, respectively.

\hat{i}, \hat{j}

and

\hat{k}

are unit vectors along the positive x, y and z-directions, respectively. If

\hat{k} = \hat{i} \times \hat{j}

then which of the following statement(s) is(are) correct ?

(A)The particle arrives at the point ( $x = l$ , $y = -h$ ) at time t = 2s.
(B) $\vec{\tau} = 2\hat{k}$ when the particle passes through the point ( $x = l$ , $y = -h$ )
(C) $\vec{L} = 4\hat{k}$ when the particle passes through the point ( $x = l$ , $y = -h$ )
(D) $\vec{\tau} = \hat{k}$ when the particle passes through the point ( $x = 0$ , $y = -h$ )

Q157·PhysicsIntegerJEE Advanced 2021

A thin rod of mass M and length

a

is free to rotate in horizontal plane about a fixed vertical axis passing through point O. A thin circular disc of mass M and of radius

a/4

is pivoted on this rod with its center at a distance

a/4

from the free end so that it can rotate freely about its vertical axis, as shown in the figure. Assume that both the rod and the disc have uniform density and they remain horizontal during the motion. An outside stationary observer finds the rod rotating with an angular velocity

\Omega

and the disc rotating about its vertical axis with angular velocity

4\Omega

. The total angular momentum of the system about the point O is

\left(\frac{Ma^2\Omega}{48}\right)n

. The value of n is____.

Q158·PhysicsSingle correctJEE Main 2021

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

Q159·PhysicsSingle correctJEE Main 2021

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) $18.75$ kgm $^2$
(B) $1.905 \times 10^5$ kgm $^2$
(C) $19.05$ kgm $^2$
(D) $1.875 \times 10^5$ kgm $^2$

Q160·PhysicsSingle correctJEE Main 2021

Two discs have moments of intertia

I_{1}

and

I_{2}

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) $\frac{I_{1}I_{2}}{(I_{1}+I_{2})}\left(\omega_{1}-\omega_{2}\right)^{2}$
(B) $\frac{(I_{1}-I_{2})^{2}\,\omega_{1}\omega_{2}}{2(I_{1}+I_{2})}$
(C) $\frac{I_{1}I_{2}}{2(I_{1}+I_{2})}\left(\omega_{1}-\omega_{2}\right)^{2}$
(D) $\frac{(\omega_{1}-\omega_{2})^{2}}{2(I_{1}+I_{2})}$

Q161·PhysicsSingle correctJEE Main 2021

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) $\frac{Ml^2}{6}$
(B) $Ml^2$
(C) $\frac{Ml^2}{12}$
(D) $\frac{2}{3}Ml^2$

Q162·PhysicsSingle correctJEE Main 2021

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 m

^{2}

.

(A)14.9 kg / m $^{3}$
(B) $7.5 \times 10^{1}$ kg / m $^{3}$
(C) $7.5 \times 10^{2}$ kg/m $^{3}$
(D) $1.49 \times 10^{2}$ kg / m $^{3}$

Q163·PhysicsNumericalJEE Main 2021

Consider a badminton racket with length scales as shown in the figure. If the mass of the linear and circular portions of the badminton racket are same (M) and the mass of the threads are negligible, the moment of inertia of the racket about an axis perpendicular to the handle and in the plane of the ring at,

\dfrac{r}{2}

distance from the end A of the handle will be ........

\mathrm{Mr^2}

.

Q164·PhysicsNumericalJEE Main 2021

In the given figure, two wheels P and Q are connected by a belt B. The radius of P is three times as that of Q. In case of same rotational kinetic energy, the ratio of rotational inertias

\left(\frac{I_1}{I_2}\right)

will be x : 1. The value of x will be __________.

Q165·PhysicsSingle correctJEE Main 2021

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 ⊥ to the plane of the disc and passing through its centre) to 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) $R^2 : r^2$
(B) $2r^4 : R^4$
(C) $2R^4 : r^4$
(D) $2R^2 : r^2$

Q166·PhysicsNumericalJEE Main 2021

A solid disc of radius 20 cm and mass 10 kg is rotating with an angular velocity of 600 rpm, about an axis normal to its circular plane and passing through its centre of mass. The retarding torque required to bring the disc at rest in 10s is ________

\pi \times 10^{-1}

Nm.

Q167·PhysicsNumericalJEE Main 2021

A particle of mass 'm' is moving in time 't' on a trajectory given by

\vec{r} = 10\alpha t^2\hat{i} + 5\beta\left(t - 5\right)\hat{j}

Where α and β are dimensional constants. The angular momentum of the particle become the same as it was for t = 0 at time t = ________seconds.

Q168·PhysicsSingle correctJEE Main 2021

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 '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

&

I_z

respectively. The respective 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)A is correct but R is not correct .
(B)Both A and B are correct and R is the correct explanation of A .
(C)Both A and R are correct but R is NOT the correct explanation of A.
(D)A is not correct but R is correct.

Q169·PhysicsSingle correctJEE Main 2021

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)The ring has the greatest and the cylinder has the least velocity of the centre of mass at the bottom of the inclined plane.
(B)The sphere has the greatest and the ring has the least velocity of the centre of mass at the bottom of the inclined plane.
(C)The cylinder has the greatest and the sphere has the least velocity of the centre of mass at the bottom of the inclined plane.
(D)All of them will have same velocity.

Q170·PhysicsNumericalJEE Main 2021

The centre of a wheel rolling on a plane surface moves with a speed

\upsilon_0

. A particle on the rim of the wheel at the same level as the centre will be moving at a speed

\sqrt{x}\,\upsilon_0

. Then the value of x is ________.

Q171·PhysicsNumericalJEE Main 2021

The position of the centre of mass of a uniform semi − circular wire of radius 'R' placed in x − y plane with its centre at the origin and the line joining its ends as x-axis is given by

\left(0, \frac{xR}{\pi}\right)

. Then, the value of |x| is ________.

Q172·PhysicsNumericalJEE Main 2021

A rod of mass M and length L is lying on a horizontal frictionless surface. A particle of mass 'm' travelling along the surface hits at one end of the rod with a velocity 'u' in a direction perpendicular to the rod. The collision is completely elastic. After collision, particle comes to rest. The ratio of masses

\left( \frac{m}{M} \right)

is

\frac{1}{x}

. The value of 'x' will be ________.

Q173·PhysicsNumericalJEE Main 2021

Two bodies, a ring and a solid cylinder of same material are rolling down without slipping an inclined plane. The radii of the bodies are same. The ratio of velocity centre of mass at the bottom of the inclined plane of the ring to that of the cylinder is

\frac{\sqrt{x}}{2}

. Then, the value of x is ___________.

Q174·PhysicsNumericalJEE Main 2021

A body rotating with an angular speed of 600 rpm is uniformly accelerated to 1800 rpm in 10 sec. The number of rotations made in the process is ____________.

Q175·PhysicsSingle correctJEE Main 2021

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)Hollow cylinder
(B)Ring
(C)Solid sphere
(D)Solid cylinder

Q176·PhysicsSingle correctJEE Main 2021

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 centre is :

(A) $\frac{1}{4}\frac{ML^{2}}{\pi^{2}}$
(B) $\frac{2}{5}\frac{ML^{2}}{\pi^{2}}$
(C) $\frac{ML^{2}}{\pi^{2}}$
(D) $\frac{1}{2}\frac{ML^{2}}{\pi^{2}}$

Q177·PhysicsSingle correctJEE Main 2021

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

(A) $\omega\dfrac{M}{M+m}$
(B) $\omega\dfrac{M+2m}{M}$
(C) $\omega\dfrac{M}{M+2m}$
(D) $\omega\dfrac{M-2m}{M+2m}$

Q178·PhysicsNumericalJEE Main 2021

The following bodies, (1) a ring (2) a disc (3) a solid cylinder (4) a solid sphere, of same mass 'm' and radius 'R' are allowed to roll down without slipping simultaneously from the top of the inclined plane. The body which will reach first at the bottom of the inclined plane is ______. [Mark the body as per their respective numbering given in the question]

Q179·PhysicsSingle correctJEE Main 2021

A sphere of mass 2kg 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° with the horizontal plane, without slipping. How low will the sphere take to return to the starting point A ?

(A)0.60 s
(B)0.52 s
(C)0.57 s
(D)0.80 s

Q180·PhysicsSingle correctJEE Main 2021

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 B is

L_B

. The correct statement for this system is :

(A) $L_A$ and $L_B$ are both constant 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) $L_A$ is constant, both in magnitude and direction

Q181·PhysicsNumericalJEE Main 2021

The disc of mass M with uniform surface mass density σ is shown in the figure. The centre of mass of the quarter disc (the shaded area) is at the position

\frac{x}{3}\frac{a}{\pi}

,

\frac{x}{3}\frac{a}{\pi}

where x is ________. (Round off to the Nearest Integer) [a is an area as shown in the figure]

Q182·PhysicsNumericalJEE Main 2021

The angular speed of truck wheel is increased from 900 rpm to 2460 rpm in 26 seconds. The number of revolutions by the truck engine during this time is _______. (Assuming the acceleration to be uniform).

Q183·PhysicsSingle correctJEE Main 2021

A triangular plate is shown. A force

\vec{F} = 4\hat{i} - 3\hat{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) $-15 + 20\sqrt{3}$ , $15 + 20\sqrt{3}$

Q184·PhysicsNumericalJEE Main 2021

A force

\vec{F} = 4\hat{i} + 3\hat{j} + 4\hat{k}

is applied on an intersection point of x = 2 plane and x-axis. The magnitude of torque of this force about a point (2, 3, 4) is ________. (Round off to the Nearest Integer)

Q185·PhysicsNumericalJEE Main 2021

Consider a 20 kg uniform circular disk of radius 0.2 m. It is pin supported at its center and is at rest initially. The disk is acted upon by a constant force F = 20 N through a massless string wrapped around its periphery as shown in the figure. Suppose the disk makes n number of revolutions to attain an angular speed of 50 rad s

^{-1}

. The value of n, to the nearest integer, is _______. [Given : In one complete revolution, the disk rotates by 6.28 rad]

Q186·PhysicsNumericalJEE Main 2021

A solid disc of radius 'a' and mass 'm' rolls down without slipping on an inclined plane making an angle θ with the horizontal. The acceleration of the disc will be

\frac{2}{b}g\sin\theta

where b is ______. (Round off to the Nearest Integer) (g = acceleration due to gravity) (θ = angle as shown in figure)

Q187·PhysicsSingle correctJEE Main 2021

Four equal masses, m each are placed at the corners 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 :

(A) $ml^{2}$
(B) $2\,ml^{2}$
(C) $3\,ml^{2}$
(D) $\sqrt{3}\,ml^{2}$

Q188·PhysicsSingle correctJEE Main 2021

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) $\frac{2gh}{I+mr^{2}}$
(B)2gh
(C) $\frac{2mgh}{I+2mr^{2}}$
(D) $\frac{2mgh}{I+mr^{2}}$

Q189·PhysicsSingle correctJEE Main 2021

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) $\frac{4}{5}$ ma $^{2}$
(B) $\frac{8}{5}$ ma $^{2}$ + mb $^{2}$
(C) $\frac{4}{5}$ ma $^{2}$ + 2mb $^{2}$
(D) $\frac{8}{5}$ ma $^{2}$ + 2mb $^{2}$

Q190·PhysicsSingle correctJEE Main 2021

A circular hole of radius

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

is cut out of a circular disc of radius 'a' shown in figure. The centroid of the remaining circular portion with respect to point 'O' will be :

(A) $\frac{10}{11}a$
(B) $\frac{2}{3}a$
(C) $\frac{1}{6}a$
(D) $\frac{5}{6}a$

Q191·PhysicsSingle correctJEE Main 2021

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

I_1

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

I_2

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

I_3

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

I_4

= M.I. of solid sphere about its diameter. Then :-

(A) $I_1 = I_2 = I_3 < I_4$
(B) $I_1 + I_2 = I_3 + \frac{5}{2}I_4$
(C) $I_1 + I_3 < I_2 + I_4$
(D) $I_1 = I_2 = I_3 > I_4$

Q192·PhysicsNumericalJEE Main 2021

A uniform thin bar of mass 6 kg and length 2.4 meter is bent to make an equilateral hexagon. The moment of inertia about an axis passing through the centre of mass and perpendicular to the plane of hexagon is _______

\times 10^{-1}

kg m

^{2}

.

Q193·PhysicsSingle correctJEE Advanced 2020

A small roller of diameter 20 cm has an axle of diameter 10 cm (see figure below on the left). It is on a horizontal floor and a meter scale is positioned horizontally on its axle with one edge of the scale on top of the axle (see figure on the right). The scale is now pushed slowly on the axle so that it moves without slipping on the axle, and the roller starts rolling without slipping. After the roller has moved 50 cm, the position of the scale will look like (figures are schematic and not drawn to scale)-

(A)(A)
(B)(B)
(C)(C)
(D)(D)

Q194·PhysicsSingle correctJEE Advanced 2020

A football of radius R is kept on a hole of radius r

(r < R)

made on a plank kept horizontally. One end of the plank is now lifted so that it gets tilted making an angle

\theta

from the horizontal as shown in the figure below. The maximum value of

\theta

so that the football does not start rolling down the plank satisfies (figure is schematic and not drawn to scale) -

(A) $\sin\theta = \frac{r}{R}$
(B) $\tan\theta = \frac{r}{R}$
(C) $\sin\theta = \frac{r}{2R}$
(D) $\cos\theta = \frac{r}{2R}$

Q195·PhysicsMultiple correctJEE Advanced 2020

A rod of mass

m

and length

L

, pivoted at one of its ends, is hanging vertically. A bullet of the same mass moving at speed

v

strikes the rod horizontally at a distance

x

from its pivoted end and gets embedded in it. The combined system now rotates with angular speed

\omega

about the pivot. The maximum angular speed

\omega_M

is achieved for

x = x_M

. Then

(A) $\omega = \frac{3vx}{L^2 + 3x^2}$
(B) $\omega = \frac{12vx}{L^2 + 12x^2}$
(C) $x_M = \frac{L}{\sqrt{3}}$
(D) $\omega_M = \frac{v}{2L}\sqrt{3}$

Q196·PhysicsSingle correctJEE Main 2020

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) $ml^{2}\omega$
(B) $5ml^{2}\omega$
(C) $3ml^{2}\omega$
(D) $2ml^{2}\omega$

Q197·PhysicsSingle correctJEE Main 2020

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) $\dfrac{MR^{2}}{2}$
(B) $\dfrac{M\left(R^{2}+H^{2}\right)}{4}$
(C) $\dfrac{MH^{2}}{3}$
(D) $\dfrac{MR^{2}}{3}$

Q198·PhysicsSingle correctJEE Main 2020

The liner mass density of a thin rod AB of length L varies from A to B as

\lambda(x) = \lambda_{0}\left(1+\frac{x}{L}\right)

, where x is the distance from A. If M is the mass of the rod hen its moment of inertia about an axis passing through A and perpendicular to the rod is:

(A) $\frac{5}{12}ML^{2}$
(B) $\frac{7}{18}ML^{2}$
(C) $\frac{2}{5}ML^{2}$
(D) $\frac{3}{7}ML^{2}$

Q199·PhysicsNumericalJEE Main 2020

The centre of mass of a solid hemisphere of radius 8 cm is x cm from the centre of the flat surface. Then value of x is ______.

Q200·PhysicsNumericalJEE Main 2020

A force

\vec{F} = (\hat{i} + 2\hat{j} + 3\hat{k})

N acts at a point

(4\hat{i} + 3\hat{j} - \hat{k})

m. Then the magnitude of torque about the point

(\hat{i} + 2\hat{j} + \hat{k})

m will be

\sqrt{x}

N-m. The value of x is ________.

Q201·PhysicsSingle correctJEE Main 2020

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) $\frac{3}{4}$
(B) $\frac{5}{6}$
(C) $\frac{1}{4}$
(D)0

Q202·PhysicsNumericalJEE Main 2020

A thin rod of mass 0.9 kg and length 1 m is suspended, at rest, from one end so that it can freely oscillate in the vertical plane. A particle of move 0.1 kg moving in a straight line with velocity 80 m/s hits the rod at its bottom most point and sticks to it (see figure). The angular speed (in rad/s) of the rod immediately after the collision will be __________.

Q203·PhysicsSingle correctJEE Main 2020

Consider two uniform discs of the same thickness and different radii

R_1 = R

and

R_2 = \alpha R

made of the same material. If the ratio of their moments of inertia

I_1

and

I_2

, respectively, about their axes is

I_1 : I_2 = 1 : 16

then the value of α is:

(A)4
(B) $\sqrt{2}$
(C)2
(D) $2\sqrt{2}$

Q204·PhysicsSingle correctJEE Main 2020

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)1/2
(B)2/3
(C)1/4
(D)1/8

Q205·PhysicsNumericalJEE Main 2020

ABC is a plane lamina of the shape of an equilateral triangle. D, E are mid points of AB, AC and G is the centroid of the lamina. Moment of inertia of the lamina about an axis passing through G and perpendicular to the plane ABC is

I_0

. If part ADE is removed, the moment of inertia of the remaining part about the same axis is

\frac{NI_0}{16}

where N is an integer. Value of N is _________.

Q206·PhysicsNumericalJEE Main 2020

A circular disc of mass M and radius R is rotating about its axis with angular speed

\omega_1

. If another stationary disc having radius

\frac{R}{2}

and same mass M is dropped co-axially on to the rotating disc. Gradually both discs attain constant angular speed

\omega_2

. The energy lost in the process is p% of the initial energy. Value of p is ________.

Q207·PhysicsSingle correctJEE Main 2020

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 its as shown. The rod is pivoted about O and swings as a result of the collision making angle θ before momentarily coming to rest. If the rod has mass M = 2 kg, and length

\ell

= 1 m, the value of θ is approximately: (take g = 10 m/s

^{2}

)

(A)49°
(B)55°
(C)69°
(D)63°

Q208·PhysicsSingle correctJEE Main 2020

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 ω the rod makes an angle θ with it (see figure). To find θ equate the rate of change of angular momentum (direction going into the paper)

\dfrac{ml^{2}}{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 θ is then such that:

(A) $\cos\theta = \dfrac{g}{l\omega^{2}}$
(B) $\cos\theta = \dfrac{2g}{3l\omega^{2}}$
(C) $\cos\theta = \dfrac{3g}{2l\omega^{2}}$
(D) $\cos\theta = \dfrac{g}{2l\omega^{2}}$

Q209·PhysicsNumericalJEE Main 2020

A person of 80 kg mass is standing on the rim of a circular platform of mass 200 kg rotating about its axis at 5 revolutions per minute (rpm). The person now starts moving towards the centre of the platform. What will be the rotational speed (in rpm) of the platform when the person reaches its centre __________.

Q210·PhysicsNumericalJEE Main 2020

A massless equilateral triangle EFG of side ‘a’ (As shown in figure) has three particles of mass m situated at its vertices. The moment of inertia of the system about the line EX perpendicular to EG in the plane of EFG is

\frac{N}{20}ma^{2}

where N is an integer. The value of N is __________.

Q211·PhysicsSingle correctJEE Main 2020

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(\dfrac{R^{2}}{4} + \dfrac{L^{2}}{12}\right)

. If such a cylinder is to be made for a given mass of a material, the ratio L/R for it to have minimum possible I is

(A) $\dfrac{2}{3}$
(B) $\sqrt{\dfrac{3}{2}}$
(C) $\dfrac{3}{2}$
(D) $\sqrt{\dfrac{2}{3}}$

Q212·PhysicsSingle correctJEE Main 2020

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

I_{0}/I_{A}

of moment of inertia

I_{0}

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) $\dfrac{15}{13}$
(B) $\dfrac{13}{15}$
(C) $\dfrac{23}{13}$
(D) $\dfrac{13}{23}$

Q213·PhysicsNumericalJEE Main 2020

One end of a straight uniform 1 m long bar is pivoted on horizontal table. It is released from rest when it makes an angle

30^{\circ}

from the horizontal (see figure). Its angular speed when it hits the table is given as

\sqrt{n}

s

^{-1}

, where n is an integer. The value of n is ____.

Q214·PhysicsSingle correctJEE Main 2020

A rod of length L has non-uniform linear mass density given by

\rho(x) = a + b\left(\frac{x}{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:

(A) $\frac{3}{2}\left(\frac{a + b}{2a + b}\right)L$
(B) $\frac{4}{3}\left(\frac{a + b}{2a + 3b}\right)L$
(C) $\frac{3}{4}\left(\frac{2a + b}{3a + b}\right)L$
(D) $\frac{3}{2}\left(\frac{2a + b}{3a + b}\right)L$

Q215·PhysicsSingle correctJEE Main 2020

Consider a uniform rod of mass M = 4 m and length

\ell

pivoted about its centre. A mass m moving with velocity v making angle

\theta = \dfrac{\pi}{4}

to the rod's long axis collides with one end of the rod and sticks to it. The angular sped of the rod-mass system just after the collision is

(A) $\dfrac{3}{7}\dfrac{v}{\ell}$
(B) $\dfrac{3\sqrt{2}}{7}\dfrac{v}{\ell}$
(C) $\dfrac{4}{7}\dfrac{v}{\ell}$
(D) $\dfrac{3}{7\sqrt{2}}\dfrac{v}{\ell}$

Q216·PhysicsSingle correctJEE Main 2020

As shown in figure when a spherical cavity (centered at O) of radius 1 is cut out of a uniform sphere of radius R (centred 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 determined by the equation:

(A)(R $^{2}$ + R + 1) (2 $-$ R) = 1
(B)(R $^{2}$ + R $-$ 1) (2 $-$ R) = 1
(C)(R $^{2}$ $-$ R + 1) (2 $-$ R) = 1
(D)(R $^{2}$ $-$ R $-$ 1) (2 $-$ R) = 1

Q217·PhysicsSingle correctJEE Main 2020

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

(A)(0.75 m, 0.75 m)
(B)(0.75 m, 1.75 m)
(C)(1.25 m, 1.50 m)
(D)(1 m, 1.75 m)

Q218·PhysicsSingle correctJEE Main 2020

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) $8.75\times 10^{-4}$ J
(B) $8.75\times 10^{-3}$ J
(C) $6.25\times 10^{-4}$ J
(D) $1.13\times 10^{-3}$ J

Q219·PhysicsSingle correctJEE Main 2020

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) $r\sqrt{\dfrac{3}{4gh}}$
(B) $\dfrac{1}{r}\sqrt{\dfrac{2gh}{3}}$
(C) $r\sqrt{\dfrac{3}{2gh}}$
(D) $\dfrac{1}{r}\sqrt{\dfrac{4gh}{3}}$

Q220·PhysicsSingle correctJEE Main 2020

Three point particles of masses 10 kg, 1.5 kg and 2.5 kg are placed at there corners of a right angle triangle of sides 4.0 cm, 3.0 ;cm and 5.0 cm as shown in the figure. The centre of mass of the system is at a point:

(A)0.6 cm right and 2.0 cm above 1 kg mass.
(B)2.0 cm right and 0.9 cm above 1 kg mass.
(C)1.5 cm right and 1.2 cm above 1 kg mass.
(D)0.9 cm right and 2.0 cm above 1 kg mass.

Q221·PhysicsNumericalJEE Main 2020

Consider a uniform cubical box of side a on a rough floor that is to be moved by applying minimum possible force F at a point b above its centre of mass (see figure). If the coefficient of friction is

\mu

= 0.4, the maximum possible value of

100 \times \dfrac{b}{a}

for box not to topple before moving is __________.

Q222·PhysicsSingle correctJEE Main 2020

Mass per unit are of a circular disc of radius a depends on the distance r from its centre as

\sigma(r) = A + Br

. The moment of inertia of the disc about the axis, perpendicular to the plane and passing through its centre is:

(A) $2\pi a^{4}\left(\dfrac{A}{4}+\dfrac{B}{5}\right)$
(B) $2\pi a^{4}\left(\dfrac{A}{4}+\dfrac{aB}{5}\right)$
(C) $2\pi a^{4}\left(\dfrac{aA}{4}+\dfrac{B}{5}\right)$
(D) $\pi a^{4}\left(\dfrac{A}{4}+\dfrac{aB}{5}\right)$

Q223·PhysicsSingle correctJEE Main 2020

The radius of gyration of a uniform rod of length

\ell

, about an axis passing through a point

\dfrac{\ell}{4}

away from the centre of the rod, an perpendicular to it is:

(A) $\dfrac{1}{8}\ell$
(B) $\sqrt{\dfrac{7}{48}}\ell$
(C) $\dfrac{1}{4}\ell$
(D) $\sqrt{\dfrac{3}{8}}\ell$

Q224·PhysicsMultiple correctJEE Advanced 2019

A thin and uniform rod of mass M and length L is held vertical on a floor with large friction. The rod is released from rest so that it falls by rotating about its contact-point with the floor without slipping. Which of the following statement(s) is/are correct, when the rod makes an angle

60^\circ

with vertical ? [g is the acceleration due to gravity]

(A)The radial acceleration of the rod's center of mass will be $\frac{3g}{4}$
(B)The angular speed of the rod will be $\sqrt{\frac{3g}{2L}}$
(C)The angular acceleration of the rod will be $\frac{2g}{L}$
(D)The normal reaction force from the floor on the rod will be $\frac{Mg}{16}$

Q225·PhysicsSingle correctJEE Main 2019

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 :

(A) $\left(\dfrac{\sqrt{3}}{7} m, \dfrac{7}{12} m\right)$
(B) $\left(\dfrac{7}{12} m, \dfrac{\sqrt{3}}{8} m\right)$
(C) $\left(\dfrac{\sqrt{3}}{4} m, \dfrac{5}{12} m\right)$
(D) $\left(\dfrac{7}{12} m, \dfrac{\sqrt{3}}{4} m\right)$

Q226·PhysicsSingle correctJEE Main 2019

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(\dfrac{\sigma_0}{r}\right)

, then the radius of gyration of the disc about its axis passing through the centre is :

(A) $\dfrac{a+b}{3}$
(B) $\sqrt{\dfrac{a^{2}+b^{2}+ab}{3}}$
(C) $\dfrac{a+b}{2}$
(D) $\sqrt{\dfrac{a^{2}+b^{2}+ab}{2}}$

Q227·PhysicsSingle correctJEE Main 2019

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)(A)
(B)(B)
(C)(C)
(D)(D)

Q228·PhysicsSingle correctJEE Main 2019

A particle of mass m is moving along a trajectory given by

x = x_{0} + a\cos\omega_{1}t

y = y_{0} + b\sin\omega_{2}t

The torque, acing on the particle about the origin, at t = 0 is:

(A) $my_{0}a\omega_{1}^{2}\hat{k}$
(B) $m(-x_{0}b + y_{0}a)\omega_{1}^{2}\hat{k}$
(C) $-m(-x_{0}b\omega_{2}^{2} - y_{0}a\omega_{1}^{2})\hat{k}$
(D)Zero

Q229·PhysicsSingle correctJEE Main 2019

The time dependence of the position of a particle of mass m = 2 is given by

\vec{r}(t) = 2t\hat{i} - 3t^{2}\hat{j}

Its angular momentum with respect to the origin at time t = 2 is .

(A) $-48\hat{k}$
(B) $48(\hat{i} + \hat{j})$
(C) $36\hat{k}$
(D) $-34(\hat{k} - \hat{i})$

Q230·PhysicsSingle correctJEE Main 2019

Two coaxial discs, having moments of inertia I

_{1}

and

\dfrac{I_{1}}{2}

, area rotating with respectively angular velocities

\omega_{1}

and

\dfrac{\omega_{1}}{2}

, about their common axes. 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) $\dfrac{I_{1}\omega_{1}^{2}}{6}$
(B) $\dfrac{3}{8}I_{1}\omega_{1}^{2}$
(C) $-\dfrac{I_{1}\omega_{1}^{2}}{12}$
(D) $-\dfrac{I_{1}\omega_{1}^{2}}{24}$

Q231·PhysicsSingle correctJEE Main 2019

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 is 5 s is close to

(A) $2.0 \times 10^{-5}$ Nm
(B) $4.0 \times 10^{-6}$ Nm
(C) $1.6 \times 10^{-5}$ Nm
(D) $7.9 \times 10^{-6}$ Nm

Q232·PhysicsSingle correctJEE Main 2019

A thin disc of mass M and radius R has mass per unit area

\sigma(r) = kr^{2}

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) $\dfrac{MR^{2}}{2}$
(B) $\dfrac{MR^{2}}{3}$
(C) $\dfrac{MR^{2}}{6}$
(D) $\dfrac{2MR^{2}}{3}$

Q233·PhysicsSingle correctJEE Main 2019

A solid sphere of mass M and radius R is divided into two unequal parts. The first part has a mass of

\dfrac{7M}{8}

and is converted into a uniform disc of radius 2R. The second part is converted into a uniform solid sphere. Let

I_{1}

be the moment of inertia of the disc about its axis and

I_{2}

be the moment of inertia of the new sphere about its axis. The ratio of

I_{1}/I_{2}

is given by:

(A)285
(B)185
(C)65
(D)140

Q234·PhysicsSingle correctJEE Main 2019

Moment of inertia of a body about a given axis is 1.5 kg m

^{2}

Initially the body is at rest. In order to produce a rotational kinetic energy of 1200 J, the angular acceleration of 20 rad/s

^{2}

must be applied about the axis of rotation for a duration of:

(A)2 s
(B)5 s
(C)2.5 s
(D)3 s

Q235·PhysicsSingle correctJEE Main 2019

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 centre. Two beads of mass m and negligible size are at the centre 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) $\dfrac{M\omega_{0}}{M + 3m}$
(B) $\dfrac{M\omega_{0}}{M + m}$
(C) $\dfrac{M\omega_{0}}{M + 2m}$
(D) $\dfrac{M\omega_{0}}{M + 6m}$

Q236·PhysicsSingle correctJEE Main 2019

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) $\dfrac{k}{I}\theta$
(B) $\dfrac{k}{2I}\theta$
(C) $\dfrac{k}{4I}\theta$
(D) $\dfrac{2k}{I}\theta$

Q237·PhysicsSingle correctJEE Main 2019

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 radio

\frac{h_{sph}}{h_{cyl}}

is given by:

(A)1
(B) $\frac{4}{5}$
(C) $\frac{2}{\sqrt{5}}$
(D) $\frac{14}{15}$

Q238·PhysicsSingle correctJEE Main 2019

A thin circular plate of mass M and radius R has its density varying as

p(r)=p_0 r

with

P_0

as constant and r is the distance from its center. The moment of Inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is

I=aMR^{2}

. The value of the coefficient a is:

(A) $\dfrac{8}{5}$
(B) $\dfrac{1}{2}$
(C) $\dfrac{3}{5}$
(D) $\dfrac{3}{2}$

Q239·PhysicsSingle correctJEE Main 2019

A uniform rectangular thin sheet ABCD of mass M has length a and breadth b, as shown in the figure. If the shaded portion HBGO is cut off, the coordinates of the centre of mass of the remaining portion will be:

(A) $\left(\dfrac{5a}{3}, \dfrac{5b}{3}\right)$
(B) $\left(\dfrac{2a}{3}, \dfrac{2b}{3}\right)$
(C) $\left(\dfrac{3a}{4}, \dfrac{3b}{4}\right)$
(D) $\left(\dfrac{5a}{12}, \dfrac{5b}{12}\right)$

Q240·PhysicsSingle correctJEE Main 2019

Four particles A, B, C and D with masses

m_A = m

,

m_B = 2m

,

m_c = 3m

and

m_D = 4m

are at the corners of a square. They have accelerations of equal magnitude with directions as shown. The acceleration of the centre of mass of the particles is:

(A) $\dfrac{a}{5}(\hat{i} - \hat{j})$
(B)Zero
(C) $\dfrac{a}{5}(\hat{i} + \hat{j})$
(D) $a(\hat{i} + \hat{j})$

Q241·PhysicsSingle correctJEE Main 2019

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/s

^{2}

]

(A)2 kg-m $^{2}$ /s
(B)8 kg-m $^{2}$ /s
(C)6 kg-m $^{2}$ /s
(D)3 kg-m $^{2}$ /s

Q242·PhysicsSingle correctJEE Main 2019

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)12 cm
(B)16 cm
(C)14 cm
(D)18 cm

Q243·PhysicsSingle correctJEE Main 2019

The position vector of the centre of mass

\vec{r}

cm of an asymmetric uniform bar of negligible area of cross-section as shown in figure is:

(A) $\vec{r}$ cm $= \frac{13}{8}L\hat{x} + \frac{5}{8}L\hat{y}$
(B) $\vec{r}$ cm $= \frac{5}{8}L\hat{x} + \frac{13}{8}L\hat{y}$
(C) $\vec{r}$ cm $= \frac{3}{8}L\hat{x} + \frac{11}{8}L\hat{y}$
(D) $\vec{r}$ cm $= \frac{11}{8}L\hat{x} + \frac{3}{8}L\hat{y}$

Q244·PhysicsSingle correctJEE Main 2019

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)(A)
(B)(B)
(C)(C)
(D)(D)

Q245·PhysicsSingle correctJEE Main 2019

A circular disc D

_1

of mass M and radius R has two identical discs D

_2

and D

_3

of the same mass M and radius R attached rigidly as its opposite ends (see figure). The moment of inertia of the system about the axis OO', passing through the centre of D

_1

as shown in the figure, will :

(A)MR $^{2}$
(B)3MR $^{2}$
(C) $\frac{4}{5}$ MR $^{2}$
(D) $\frac{2}{3}$ MR $^{2}$

Q246·PhysicsSingle correctJEE Main 2019

The 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 I

_0

. 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)I = $\frac{15}{16}$ I $_0$
(B)I = $\frac{3}{4}$ I $_0$
(C)I = $\frac{9}{16}$ I $_0$
(D)I = $\frac{I_0}{4}$

Q247·PhysicsSingle correctJEE Main 2019

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) $20\text{ rad/s}^{2}$
(B) $16\text{ rad/s}^{2}$
(C) $12\text{ rad/s}^{2}$
(D) $10\text{ rad/s}^{2}$

Q248·PhysicsSingle correctJEE Main 2019

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

(A) $\frac{\pi}{6}$
(B) $\frac{\pi}{3}$
(C) $\frac{\pi}{8}$
(D) $\frac{\pi}{4}$

Q249·PhysicsSingle correctJEE Main 2019

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

, then:

(A) $\tan \theta = \dfrac{1}{2\sqrt{3}}$
(B) $\tan \theta = \dfrac{1}{2}$
(C) $\tan \theta = \dfrac{2}{\sqrt{3}}$
(D) $\tan \theta = \dfrac{1}{3}$

Q250·PhysicsSingle correctJEE Main 2019

A rod of length 50 cm is pivoted at one end. It is raised such that if makes an angle of 30° fro 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) $\sqrt{\dfrac{30}{2}}$
(B) $\sqrt{30}$
(C) $\sqrt{\dfrac{20}{2}}$
(D) $\dfrac{\sqrt{30}}{2}$

Q251·PhysicsMultiple correctJEE Advanced 2018

Consider a body of mass

1.0\ kg

at rest at the origin at time

t = 0

. A force

\vec{F} = \left(\alpha t\,\hat{i} + \beta\,\hat{j}\right)

is applied on the body, where

\alpha = 1.0\,Ns^{-1}

and

\beta = 1.0\,N

. The torque acting on the body about the origin at time

t = 1.0\ s

is

\vec{\tau}

. Which of the following statements is (are) true?

(A) $\left|\vec{\tau}\right| = \frac{1}{3}\,Nm$
(B)The torque $\vec{\tau}$ is in the direction of the unit vector $+\ \hat{k}$
(C)The velocity of the body at $t = 1s$ is $\vec{v} = \frac{1}{2}\left(\hat{i} + 2\hat{j}\right)ms^{-1}$
(D)The magnitude of displacement of the body at $t = 1\ s$ is $\frac{1}{6}\,m$

Q252·PhysicsNumericalJEE Advanced 2018

A ring and a disc are initially at rest, side by side, at the top of an inclined plane which makes an angle

60^{\circ}

with the horizontal. They start to roll without slipping at the same instant of time along the shortest path. If the time difference between their reaching the ground is

(2 - \sqrt{3})/\sqrt{10}\ s

, then the height of the top of the inclined plane, in metres, is __________. Take

g = 10\ ms^{-2}

.

Q253·PhysicsSingle correctJEE Advanced 2017

PARAGRAPH 2 One twirls a circular ring (of mass M and radius R) near the tip of one's finger as shown in Figure 1. In the process the finger never loses contact with the inner rim of the ring. The finger traces out the surface of a cone, shown by the dotted line. The radius of the path traced out by the point where the ring and the finger is in contact is r. The finger rotates with an angular velocity

\omega_{0}

. The rotating ring rolls without slipping on the outside of a smaller circle described by the point where the ring and the finger is in contact (Figure 2). The coefficient of friction between the ring and the finger is

\mu

and the acceleration due to gravity is g. The total kinetic energy of the ring is

(A) $M\omega_{0}^{2}R^{2}$
(B) $\dfrac{1}{2}M\omega_{0}^{2}(R-r)^{2}$
(C) $M\omega_{0}^{2}(R-r)^{2}$
(D) $\dfrac{3}{2}M\omega_{0}^{2}(R-r)^{2}$

Q254·PhysicsSingle correctJEE Advanced 2017

Consider regular polygons with number of sides

n = 3, 4, 5, \dots

as shown in the figure. The center of mass of all the polygons is at height

h

from the ground. They roll on a horizontal surface about the leading vertex without slipping and sliding as depicted. The maximum increase in height of the locus of the center of mass for each polygon is

\Delta

. Then

\Delta

depends on

n

and

h

as

(A) $\Delta = h\sin^{2}\!\left(\dfrac{\pi}{n}\right)$
(B) $\Delta = h\left(\dfrac{1}{\cos\left(\frac{\pi}{n}\right)} - 1\right)$
(C) $\Delta = h\sin\!\left(\dfrac{2\pi}{n}\right)$
(D) $\Delta = h\tan^{2}\!\left(\dfrac{\pi}{2n}\right)$

Q255·PhysicsMultiple correctJEE Advanced 2017

A block of mass M has a circular cut with a frictionless surface as shown. The block rests on the horizontal frictionless surface of a fixed table. Initially the right edge of the block is at x = 0, in a co-ordinate system fixed to the table. A point mass m is released from rest at the topmost point of the path as shown and it slides down. When the mass loses contact with the block, its position is x and the velocity is v. At that instant, which of the following options is/are correct?

(A)The x component of displacement of the center of mass of the block M is : $-\frac{mR}{M+m}$ .
(B)The position of the point mass is : $x = -\sqrt{2}\,\frac{mR}{M+m}$ .
(C)The velocity of the point mass m is : $v = \sqrt{\frac{2gR}{1+\frac{m}{M}}}$ .
(D)The velocity of the block M is: $V = -\frac{m}{M}\sqrt{2gR}$ .

Q256·PhysicsMultiple correctJEE Advanced 2017

A rigid uniform bar AB of length L is slipping from its vertical position on a frictionless floor (as shown in the figure). At some instant of time, the angle made by the bar with the vortical is

\theta

. Which of the following statements about its motion is/are correct?

(A)The midpoint of the bar will fall vertically downward
(B)The trajectory of the point A is a parabola
(C)Instantaneous torque about the point in contact with the floor is proportional to $\sin\theta$
(D)When the bar makes an angle $\theta$ with the vertical, the displacement of its midpoint from the initial position is proportional to $(1-\cos\theta)$

Q257·PhysicsSingle correctJEE Advanced 2017

PARAGRAPH 2 One twirls a circular ring (of mass M and radius R) near the tip of one's finger as shown in Figure 1. In the process the finger never loses contact with the inner rim of the ring. The finger traces out the surface of a cone, shown by the dotted line. The radius of the path traced out by the point where the ring and the finger is in contact is r. The finger rotates with an angular velocity

\omega_{0}

. The rotating ring rolls without slipping on the outside of a smaller circle described by the point where the ring and the finger is in contact (Figure 2). The coefficient of friction between the ring and the finger is

\mu

and the acceleration due to gravity is g. The minimum value of

\omega_{0}

below which the ring will drop down is

(A) $\sqrt{\dfrac{g}{\mu(R-r)}}$
(B) $\sqrt{\dfrac{2g}{\mu(R-r)}}$
(C) $\sqrt{\dfrac{3g}{2\mu(R-r)}}$
(D) $\sqrt{\dfrac{g}{2\mu(R-r)}}$

Q258·PhysicsMultiple correctJEE Advanced 2017

A wheel of radius R and mass M is placed at the bottom of a fixed step of height R as shown in the figure. A constant force is continuously applied on the surface of the wheel so that it just climbs the step without slipping. Consider the torque

\tau

about an axis normal to the plane of the paper passing through the point Q. Which of the following options is/are correct?

(A)If the force is applied at point P tangentially then $\tau$ decreases continuously as the wheel climbs
(B)If the force is applied normal to the circumference at point X then $\tau$ is constant
(C)If the force is applied normal to the circumference at point P then $\tau$ is zero
(D)If the force is applied tangentially at point S then $\tau \neq 0$ but the wheel never climbs the step

Q259·PhysicsMultiple correctJEE Advanced 2016

The position vector

\vec{r}

of a particle of mass m is given by the following equation

\vec{r}(t) = \alpha t^{3}\hat{i} + \beta t^{2}\hat{j}

, where

\alpha = 10/3\ \text{m s}^{-3}

,

\beta = 5\ \text{m s}^{-2}

and

m = 0.1

kg. At

t = 1

s, which of the following statement(s) is(are) true about the particle?

(A)The velocity $\vec{v}$ is given by $\vec{v} = \left(10\hat{i} + 10\hat{j}\right)\ \text{m s}^{-1}$
(B)The angular momentum $\vec{L}$ with respect to the origin is given by $\vec{L} = -(5/3)\hat{k}\ \text{N m s}$
(C)The force $\vec{F}$ is given by $\vec{F} = (\hat{i} + 2\hat{j})\ \text{N}$
(D)The torque $\vec{\tau}$ with respect to the origin is given by $\vec{\tau} = -(20/3)\hat{k}\ \text{N m}$

Q260·PhysicsMultiple correctJEE Advanced 2016

Two thin circular discs of mass m and 4m, having radii of a and 2a, respectively, are rigidly fixed by a massless, rigid rod of length

\ell = \sqrt{24}\,a

through their centers. This assembly is laid on a firm and flat surface, and set rolling without slipping on the surface so that the angular speed about the axis of the rod is

\omega

. The angular momentum of the entire assembly about the point 'O' is

\vec{L}

(see the figure). Which of the following statement(s) is(are) true?

(A)The magnitude of angular momentum of the assembly about its center of mass is $17\,ma^{2}\omega/2$
(B)The magnitude of the z-component of $\vec{L}$ is $55\,ma^{2}\omega$
(C)The magnitude of angular momentum of center of mass of the assembly about the point O is $81\,ma^{2}\omega$
(D)The center of mass of the assembly rotates about the z-axis with an angular speed of $\omega/5$

Q261·PhysicsIntegerJEE Advanced 2015

Two identical uniform discs roll without slipping on two different surfaces AB and CD (see figure) starting at A and C with linear speeds

v_{1}

and

v_{2}

, respectively, and always remain in contact with the surfaces. If they reach B and D with the same linear speed and

v_{1} = 3

m/s, then

v_{2}

in m/s is (

g = 10

m/s

^{2}

)

Q262·PhysicsIntegerJEE Advanced 2015

The densities of two solid spheres A and B of the same radii R vary with radial distance r as

\rho_{A}(r) = k\left(\dfrac{r}{R}\right)

and

\rho_{B}(r) = k\left(\dfrac{r}{R}\right)^{5}

, respectively, where k is a constant. The moments of inertia of the individual spheres about axes passing through their centres are

I_{A}

and

I_{B}

, respectively. If

\dfrac{I_{B}}{I_{A}} = \dfrac{n}{10}

, the value of n is

Q263·PhysicsMultiple correctJEE Advanced 2015

A ring of mass M and radius R is rotating with angular speed

\omega

about a fixed vertical axis passing through its centre O with two point masses each of mass

\dfrac{M}{8}

at rest at O. These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant the angular speed of the system is

\dfrac{8}{9}\omega

and one of the masses is at a distance of

\dfrac{3}{5}R

from O. At this instant the distance of the other mass from O is

(A) $\dfrac{2}{3}R$
(B) $\dfrac{1}{3}R$
(C) $\dfrac{3}{5}R$
(D) $\dfrac{4}{5}R$

Q264·PhysicsIntegerJEE Advanced 2014

A horizontal circular platform of radius

0.5 \text{ m}

and mass

0.45 \text{ kg}

is free to rotate about its axis. Two massless spring toy-guns, each carrying a steel ball of mass

0.05 \text{ kg}

are attached to the platform at a distance

0.25 \text{ m}

from the centre on its either sides along its diameter (see figure). Each gun simultaneously fires the balls horizontally and perpendicular to the diameter in opposite directions. After leaving the platform, the balls have horizontal speed of

9 \text{ ms}^{-1}

with respect to the ground. The rotational speed of the platform in

\text{rad s}^{-1}

after the balls leave the platform is

Q265·PhysicsIntegerJEE Advanced 2014

A uniform circular disc of mass

1.5 \text{ kg}

and radius

0.5 \text{ m}

is initially at rest on a horizontal frictionless surface. Three forces of equal magnitude

F = 0.5 \text{ N}

are applied simultaneously along the three sides of an equilateral triangle

XYZ

with its vertices on the perimeter of the disc (see figure). One second after applying the forces, the angular speed of the disc in

\text{rad s}^{-1}

is

Q266·PhysicsIntegerJEE Advanced 2013

A uniform circular disc of mass 50 kg and radius 0.4 m is rotating with an angular velocity of 10 rad s

^{-1}

about its own axis, which is vertical. Two uniform circular rings, each of mass 6.25 kg and radius 0.2 m, are gently placed symmetrically on the disc in such a manner that they are touching each other along the axis of the disc and are horizontal. Assume that the friction is large enough such that the rings are at rest relative to the disc and the system rotates about the original axis. The new angular velocity (in rad s

^{-1}

) of the system is

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