CAPS-9

PHYSICS ROTATIONAL DYNAMICS-2

TARGET : JEE- Advanced 2025
SCQ (Single Correct Type) :
1. A solid sphere of mass m and radius r is gently placed on a conveyer belt moving with
2
constant velocity V. If the coefficient of friction between the belt and sphere is , the distance
7
travelled by the centre of the sphere before it starts pure rolling is

V2 2V 2 2V 2 2V 2
(A) (B) (C) (D)
7g 49g 5g 7g

2. Two points A & B on a disc have velocities v1 & v2 at some moment. Their directions make
angles 60° and 30° respectively with the line of separation as shown in figure. The angular
velocity of disc is :

3v1 v2 v 2  v1 v2
(A) (B) (C) (D)
d 3d d d

3. A hollow smooth uniform sphere A of mass ‘m’ rolls without sliding on a smooth horizontal
surface. It collides elastically and head on with another stationary smooth solid sphere B of
the same mass m and same radius. The ratio of kinetic energy of ‘B’ to that of ‘A’ just after the
collision is :

(A) 5 : 2 (B) 1 : 1 (C) 2 : 3 (D) 3 : 2

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4. Two identical uniform rectangular blocks (with longest side L) and a solid sphere of radius R
are to be balanced at the edge of a heavy table such that the centre of the sphere remains at
the maximum possible horizontal distance from the vertical edge of the table without toppling
as indicated in the figure.

If the mass of each block is M and of the sphere is M/2, then the maximum distance x that can
be achieved is
(A) 8L/15 (B) 5L/6 (C) (3L/4 + R) (D) (7L/15 + R)
5. A small object of uniform density rolls up a curved surface with an initial velocity v. It reaches
3v 2
up to a maximum height of with respect to the initial position. The object is
4g

v


(A) ring (B) solid sphere (C) hollow sphere (D) disc
6. A ring of mass M and radius R is rotating with angular speed  about a fixed vertical axis
M
passing through its centre O with two point masses each of mass at rest at O. These
8
masses can move radially outwards along two massless rods fixed on the ring as shown in the
8
figure. At some instant the angular speed of the system is  and one of the masses is ata
9
3
distance of R from O. At this instant the distance of the other mass from O is :
5

2 1 3 4
(A) R (B) R (C) R (D) R
3 3 5 5

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7. A uniform rod of mass m, length  is placed over a smooth horizontal surface along y-axis and

is at rest as shown in figure. An impulsive force F is applied for a small time t along
x-direction at point A after this rod moves freely. The x–coordinate of end A of the rod when
the rod becomes parallel to x–axis for the first time is (initially the coordinate of centre of mass
of the rod is (0, 0)) :

       
(A) (B) 1 (C) 1  6  (D) 1  6 
12 2  12 
 2   2  

MCQ (One or more than one correct) :

8. A rigid body undergoing pure rolling encounters horizontal rigid tracks AB and BC as shown.
AB is smooth surface and BC is rough surface with  = 1. Which of the following statements
is/are correct :

µ=1
A Smooth B Rough C

(A) Angular momentum of the rigid body is conserved only about a point on the horizontal
surface.
(B) Angular momentum of the rigid body is conserved about every point in space.
(C) In part BC, there will be no frictional force on the rigid body.
(D) In part BC, frictional force will act opposite to velocity of rigid body.
9. A sphere is rolling without slipping on a fixed horizontal plane surface. In the figure, A is the
point of contact, B is the centre of the sphere and C is its topmost point. Then,


(A) VC  VA = 2 VB  VC  (B) VC  VB = VB  VA

(C) VC  VA = 2 VB  VC (D) VC  VA = 4 VB

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10. A uniform circular disc of mass M. and radius R has four particles (each of mass m) rigidly
attached to it. The disc rolls on the surface S without slipping. At the instant shown the centre
of the disc has velocity v0,
Y
v
m

m m
m

P X
O
S

 9m 3M 
(A) The kinetic energy of the disc-particle assembly is   v 02
 2 4 

ˆ  M  5m 
(B) The velocity of the centre of mass of the assembly is   v0 i
 M  4m 
3
(C) The angular momentum of the assembly about the point of contact P is (M+ 6m)
2
(Rv0(– k̂ )
(D) The maximum speed of a point on the disc with respect to S is 2v0
11. A solid sphere is in pure rolling motion on an inclined surface having inclination .


//////////////////////

(A) frictional force acting on sphere is f =  mg cos .

(B) f is dissipative force.
(C) friction will increase its angular velocity and decreases its linear velocity.
(D) If  decreases, friction will decrease.
Comprehension Type Question:
A uniform bar of length 6 a & mass 8 m lies on a smooth horizontal table. Two point masses m
& 2 m moving in the same horizontal plane with speeds 2 v andv respectively strike the bar as
shown & stick to the bar after collision.

12. Velocity of the centre of mass of the system is

v 2v
(A) (B) v (C) (D) Zero
2 3
13. Angular velocity of the rod about centre of mass of the system is
v v v v
(A) (B) (C) (D)
5a 15a 3a 10a
14. Total kinetic energy of the system, just after the collision is
3 3 3
(A) mv2 (B) mv2 (C) mv2 (D) 3 mv2
5 25 15

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Numerical based Questions :
15. A solid billiard ball of radius ‘R’ and mass ‘m’ initially at rest is given a sharp impulse by a cue,
R
held horizontally at a distance above the centre. Just after the impulse, the velocity of
2
centre of mass of the ball is v = 10 m/s. The coefficient of friction between ball and table is
1
= . The ball starts rolling without slipping t seconds after impulse is given. Find value of
2
1
(in sec–1)
t
16. A boy is pushing a ring of mass 2 kg and radius 0.5 m with a vertical stick as shown in the
figure. The stick applies a normal force of 2 N on the ring and rolls it without slipping with an
acceleration of 0.3 m/s2. The coefficient of friction between the ground and the ring is large
enough that rolling always occurs and the coefficient of friction between the stick and the ring
is (P/10). The value of P is

17. Two identical uniform discs roll without slipping on two different surfaces AB and CD (see
figure) starting at A and C with linear speeds v1 and v2, respectively, and always remain in
contact with ths surfaces. If they reach B and D with the same linear speed and v1 = 3 m/s,
then v2 in m/s is (g = 10 m/s2)

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Matrix Match Type :
18. A uniform disc of mass M and radius R lies on a fixed rough horizontal surface at time t = 0.
Initial angular velocity o of each disc (magnitude and sense of rotation) and horizontal
velocity v0 of centre of mass is shown for each situation of column-I. Match each situation in
column- with the results given in column-.
Column-I Column-II

(A) (p) The angular speed keeps on decreasing

till the disc stops slipping.

It is given that v0 = 2R0

(B) (q) After the disc stops slipping, the angular

velocity is nonzero and in clockwise sense

It is given that 2v0 = R0

(C) (r) After the disc stops slipping, the velocity

of centre of disc is towards right

It is given that v0 = 2R0

(D) (s) After the disc stops slipping, the

kinetic energy of disc is less than its initial value.

It is given that 2v0 = R0
Subjective Type Questions :
19. In the figure shown there is a fixed wedge W of inclination ‘’. A is a block, B is a disc and ‘C’
is a solid cylinder. A, B and C each has mass m. Assuming there there is no sliding anywhere
and string to be of negligible mass find :

(a) the speed of A after descending distance ‘x’.

(b) the friction force acting on the cylinder due to the wedge.

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20. Two cylindrical rollers each of mass m are used to transport a log of mass M. If a horizontal
force F acts on the log of wood find its acceleration. There is no slipping any where

21. A 2 kg sphere moving horizontally to the right with an initial velocity of 5 m/s strikes the lower
end of an 8 kg rigid rod AB. The rod is suspended from a hinge at A and is initially at rest.
Knowing that the coefficient of restitution between the rod and sphere is 0.80, determine the
angular velocity of the rod and the velocity of the sphere immediately after the impact.
///////////////////////
A

1.2m

vs

22. A rod of length R and mass M is free to rotate about a horizontal axis passing through hinge P
as in figure. First it is taken aside such that it becomes horizontal and then released. At the
lowest point the rod hits the small block B of mass m and stops. Find the ratio of masses such
that the block B completes the circular track of radius R. Neglect any friction.

P
m

23. A plank of mass m1 with a uniform sphere of mass m2 placed on it rests on a smooth
horizontal plane. A constant horizontal force F is applied to the plank. With what accelerations
will the plank and the centre of the sphere move provided there is no sliding between the
plank and the sphere ?
24. A carpet of mass 'M' made of inextensible material is rolled along its length in the form of a
cylinder of radius 'R' and is kept on a rough floor. The carpet starts unrolling without sliding on
the floor when a negligibly small push is given to it. Calculate the horizontal velocity of the axis
of the cylindrical part of the carpet when its radius reduces to R/2.

R
R/2

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25. The surface mass density (mass/area) of a circular disc of radius 'R' depends on the distance
from the centre x given as, (x) =  + x. Where  and  are positive constant find its moment
of inertia about the line perpendicular to the plane of the disc through its centre.

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