PHYSICS
Class
II IIT-JEE Achiever
Worksheet - 30
Topics
RBD - V (Angular momentum)
Set-1
Multiple choice questions with one correct alternative
1. Angular momentum of a wheel changes from 2 L to L in 3 seconds, then magnitude of torque acting on
it is
L
L
L
(A)
(B)
(C)
(D) L
5
3
2
2. Which of the following statements about angular momentum is correct?
(A) Angular momentum remains conserved
(B) Angular momentum is the product of moment of inertia and angular velocity
(C) Moment of inertia is directly proportional to angular momentum for constant angular speed
(D) All of these
3. Two discs of moment of inertia l1 and l2 and angular speeds 1 and 2 are rotating along collinear axes
passing through their c.m. and perpendicular to their plane. If the two are made to rotate combined along
the same axis the rotational KE of system will be
l +l 
(l + l )( + 2 )
(l  + l  )2
(A) 1 1 2 2
(B) 1 2 1
(C) 1 1 2 2
(D) none of (A), (B) & (C)
2(l1 + l2 )
2
2(l1 + l2 )
4. A circular disc is rotating with angular velocity . A man sitting at the edge walks towards the centre of
disc, then the angular velocity 
(A) decreases
(B) no change
(C) increases
(D) halved
5. A particle of mass m moves along line PC with velocity v as shown. What is the angular momentum of
the particle about P?
(A) mvL
(B) mvl
6. The angular momentum of a body is conserved if
(A) force acing on it is zero
(C) force acting on it is constant
(C) mvr
(D) zero
(B) torque acting on it is zero
(D) torque acting on it is constant
7. If a gymnast sitting on a rotating stool with his arms outstretched, suddenly lowers his hands,
(A) the angular velocity decreases
(B) his moment of inertia decreases
(C) the angular velocity stays constant
(D) the angular momentum increases
1
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8. A disc is rotating with angular speed . If a child sits on it, what is conserved?
(A) Linear momentum
(B) Angular momentum
(C) Kinetic energy
(D) Potential energy
9. A planet is revolving around the sun in elliptical orbit. Its closed distance from the sun is r and the
farthest distance is R. If the orbital velocity of the planet closest to the sun be v, then what is the velocity
at the farthest point?
1/2
1/2
vR
vr
r
R
(A)
(B)
(C) v  
(D) v  
R
r
R
r
10. The angular momentum of a system of a particles is not conserved
(A) when net external force acts upon the system
(B) when a net external torque acts upon the system
(C) when a net external impulse acts upon the system
(D) none of (A), (B) & (C)
11. The angular momentum of a moving body remains constant if
(A) net external force is applied
(B) net pressure is applied
(C) net external torque is applied
(D) net external torque is not applied
12. If 2 kg mass is rotating on a circular path of radius 0.8 m with angular velocity of 44 rad s−2. If radius of
path becomes 1 m. then the value of angular velocity will be
(A) 28.16 rad s−1
(B) 35.16 rad s−1
(C) 19.28 rad s−1
(D) 8.12 rad s−1
13. A ring of radius r and mass m rotates about an axis passing through its centre and perpendicular to its
plane, its kinetic energy is
1
1
(A) mr
(B) mr 2
(C) mr22
(D) mr 2 2
2
2
14. The angular speed of a body changes from 1 to 2 without applying a torque but due to change in its
moment of inertia. The ratio of radii of gyration in the two cases is
(A) 2 − 1
(B) 1 : 2
(C) 1 : 2
(D) 2 : 1
15. A particle performing uniform circular motion has angular momentum L. If its angular frequency is
doubled and its kinetic energy halved, then the new angular momentum is
L
L
(A)
(B)
(C) 2L
(D) 4L
2
4
16. Let F be the force acing on a particle having position vector r and T be the torque of this force about
the origin. Then
(A) r.T = 0 and F.T = 0
(B) r .T = 0 and F.T  0
(C) r.T  0 and F.T = 0
(D) r .T  0 and F.T  0
17. A thin circular ring of mass M and radius r is rotating about its axis with a constant angular velocity .
Four objects each of mass m are kept gently to the opposite ends of two perpendicular diameters of the
ring. The angular velocity of the ring will be
M
(M + 4m)
(M − 4m)
M
(A)
(B)
(C)
(D)
M + 4m
M
M + 4m
4m
2
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18. Consider a body, shown in figure, consisting of two identical balls, each of mass M connected by a light
rigid rod. If an impulse J = MV is imparted to the body at one of its ends, what would be its angular
velocity?
2V
V
(A)
(B)
L
L
V
V
(C)
(D)
4L
3L
19. The motion of planets in the solar system is an example of the conservation of
(A) mass
(B) linear momentum
(C) angular momentum
(D) energy
20. A particle performs uniform circular motion with an angular momentum L. If the angular frequency of
the particle is doubled and kinetic energy is halved, its angular momentum becomes
L
L
(A) 4 L
(B) 2L
(C)
(D)
2
4
21. A one kg stone attached to the end of a 60 cm chain is revolving at the rate of 3 revolutions/second. If
after 30 seconds, it is making only one revolution per second, find the mean torque.
(A) 0.45 Nm
(B) 0.35 Nm
(C) 0.25 Nm
(D) 0.15 Nm
22. A round disc of moment of inertial I2 about its axis perpendicular to its plane and passing through its
centre is placed over another disc of moment of inertia I1 rotating with an angular velocity  about the
same axis. The final angular velocity of the combination of disc is
(I + I )
I
I
(A) 
(B) 1
(C) 1 2
(D) 2
I1
I1 + I 2
I1 + I 2
23. A solid sphere is rotating in free space. If the radius of the sphere is increased keeping mass same which
one of the following will not be affected?
(A) Moment of inertia
(B) Angular momentum
(C) Angular velocity
(D) Rotational kinetic energy
24. A point mass is rotating in a plane about a fixed point. Its angular momentum is directed along
(A) the axis of rotation
(B) a diameter of the path of rotation
(C) the tangent to the orbit
(D) an axis inclined at 45 to the plane of the orbit
25. The angular speed of a body changes from 1 to 2 without applying a torque, but due to change in
moment of inertia. The ratio of radii of gyration in the two cases is
(A) 1 : 2
(B) 2 : 1
(C) 2 : 1
(D) 1 : 2
26. A thin circular ring of mass M and radius R is rotating about its axis with a constant angular speed .
Two blocks each of mass m are attached gently to the opposite ends of a diameter of the ring.
M
M + 2m
M − 2m
M

(A)
(B)
(C)
(D)



M − 2m
M + 2m
M + 2m
M − 2m
3
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27. A particle is made to move in circular path with decreasing speed. Which of the following is correct?
(A) angular momentum is constant
(B) only the direction of L is constant
(C) acceleration is always directed towards the centre
(D) particle moves in spiral path
28. A uniform rod of length ‘8a’ and mass ‘6m’ lies on a smooth horizontal surface. Two point masses ‘6 m’
and ‘2m’ moving in the same plane with speed 2v and v respectively strike the rod perpendicularly at
distances ‘6a’ and ‘2a’ from the mid-point of the rod in the opposite directions and stick to the rod. The
angular velocity of the system immediately after the collision is
6v
6v
6v
6a
(A)
(B)
(C)
(D)
32a
40a
33a
41a
29. Law of conservation of angular momentum is applicable only when
(A) restoring force is zero
(B) force is zero
(C) torque is zero
(D) torque is constant
30. The position of a particle is given by r = i + 2 j − k and its linear momentum is given by
p = 3i + 4 j − 2k . Then its angular momentum, about the origin is perpendicular to
(A) yz-plane
(B) z-axis
(C) y-axis
(D) x-axis
Set-2
Multiple choice questions with one correct alternative
31. A uniform rod AB of length 2l and mass m is turning freely in a horizontal plane about a vertical axis
3v
through A, with angular velocity
. A particle P, also having mass m, is moving with constant velocity
l
v in the same horizontal plane. The particle and the rod are moving towards each other and when AB is
perpendicular to the path of the particle, P collides with point B of the rod. If the coefficient of restitution
1
between the rod and the particle is . Find the speed of the particle after impact.
2
13
3
(A) v
(B) v
(C) 0
(D)
v
8
8
32. A uniform disc of radius r0 lies on a smooth horizontal plane. A similar disc spinning with the angular
velocity 0 is carefully lowered onto the first disc. How soon do both discs spin with the same angularvelocity if the friction coefficient between them is equal to ?
3r 
5r 
r
r
(A) t 0 = 0 0
(B) t 0 = 0 0
(C) t 0 = 0 0
(D) t 0 = 0 0
g
8g
8g
g
33. A vertically oriented uniform rod of mass M and length l can rotate about its upper end. A horizontally
flying bullet of mass m strikes the lower end of the rod and gets embedded in it as a result of which the
rod swings through an angle . Assuming that m << M, find the velocity of the flying bullet.
m 3

M 2
M 2 sin 
m 2

gl sin
gl sin  (D) v =
gl
(A) v =
(B) v =
(C) v =
gl sin  
M 2
2
m 3
m 3
2
M 3
2
4
2I(IIT Camp)(IRP)(P3)PWS30
34. In the previous problem, the momentum increment in the system “bullet-rod” during the impact and the
cause that changes the momentum.
M

M
M


gl sin  
(A) P =
(B) P =
(C) P = Mgl sin
(D) P =
gl sin
gl sin 
2
2
6
5
6
2
35. In Q.No. 33, the distance x from the upper end of the rod the bullet must strike for the momentum of the
system “bullet-rod” to remain constant during the impact.
2
L
L
(A) n = L
(B) n = L
(C) n =
(D) n =
3
3
2
36. A rod of length l forming an angle  with the horizontal strikes a frictionless floor at A with its centre of
mass velocity v0 and no angular velocity. Assuming that A the impact at A is perfectly elastic, find the
angular velocity of the rod immediately after the impact.
v 0 cos 
6v cos 
(A)  = 0
(B) 0 =
L(1 + 3cos 2 )
L
v 0 cos 
v
(C) 0 =
(D) 0 = 0
2
L(1 + 3sin )
L
37. A solid cylinder of mass m is rolling on a rough horizontal surface with velocity v0. It collides elastically
with a cubical block of same mass at rest. The height of centre of mass of both the bodies is same.
Assume that there is no friction between the cylinder and the block and coefficient of friction between all
other surfaces is , find the time after first collision at which the second collision will take place.
(A)
( 5 + 2) v
g
0
(B)
( 5 + 2) v
2g
0
(C)
2v 0
3g
(D)
( 5 + 2) v
0
3g
38. A spherical ball of radius r and mass m collides with a plank of mass M kept on a smooth horizontal
surface. Before impact, the centre of the ball has a velocity v0 and angular velocity 0 as shown. The
normal velocity is reversed with same magnitude and the ball stops rotating after the impact. Find the
distance on the plank between first two impacts of the ball. The coefficient of friction between the ball
and the plank is . Assume that plank is large enough.
4  M + m  v0 r0
(A) 
(B) 0

5 M  g
2  M + m  v0 r0
(C) 4r
(D) 

5 M  g
39. A uniform circular disc of mass M and radius a is pivoted at a point O on its circumference so that it can
rotate about the tangent at O, which is horizontal, the centre of the disc describing a vertical circle of
centre O in a plane perpendicular to the tangent. The point diametrically opposite to O is A and the disc
is just displaced from rest when A is vertically above O. Find the angular velocity of the disc when A is
vertically below O.
g
g
(A)  = 2
(B)  = 4
5a
5a
(C)  = 0
(D)  = 3
5
5a
g
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40. At this instant a particle of mass M travelling with velocity u in the opposite direction of motion of the
centre of the disc hits the disc at its centre and adheres to it. Find the angular velocity of the system
immediately before and after the impact.
1
4
4
(A)  =
(B)  =
u − 5ag (C)  =
u − 5ag
u + 5ag (D) can’t be determined
9a
9a
3a
(
)
(
(
)
)
41. A cylinder of mass M is placed on top of a block of mass 2 M. The friction coefficient between the block
and the horizontal plane is zero and the friction coefficient between the cylinder and the top surface
block is . A small body of mass M travelling with a velocity v0, hits the block horizontally such that the
collision is completely elastic. Find the maximum value of v0 for which the cylinder does start pure
rolling before it topples from the top of the block.
21 gd
(A) gd
(B)
2 11
(C)
gd
2
(D)
gd
5
42. An impulse J is applied on a ring of mass m along a line passing through its centre O. The ring is placed
on a rough horizontal surface. The linear velocity of centre of ring once it starts rolling without slipping
is
J
J
(A)
(B)
2m
m
J
J
(C)
(D)
3m
4m
43. A solid sphere and a hollow sphere of equal mass and radius are placed over a rough horizontal surface
after rotating them about their respective centre of mass with same angular velocity 0. Once the pure
rolling starts let v1 and v2 be the linear speeds of their centres of mass. Then
(A) v1 = v2
(B) v1 > v2
(C) v1 < v2
(D) data is insufficient
44. A playground merry-go-round is at rest, pivoted about a frictionless axis. A child of mass m runs along
the path tangential to the rim with speed v and jumps on to merry-go-round. If R be the radius of merrygo-round and I is the moment of inertia, then the angular velocity of the merry-go-round and the child is
mvR
mvR
I
mR 2 + I
(A)
(B)
(C)
(D)
2
mvR
mR + I
I
mvR
45. A uniform rod AB of length L having mass m are lying on a smooth table. A small particle of mass m
strikes the rod with a velocity v0 at point C, a distance x from the centre O. The particle comes to rest
after collision. The value of x, so that point A of the rod remains stationary just after collision is
L
(A)
3
L
(B)
6
L
(C)
4
L
(D)
12
6
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Multiple choice questions with one or more than one correct alternative/s
46. Two particles A and B of mass m each hit the ends of a rigid bar of mass M and length  inelastically.
The bar is kept on a smooth horizontal plane (as shown). Then
(A) the linear speed of the system (bar + particle) after the collision zero.
12 mv
(B) the linear speed of the system (bar + particle) after the collision
(M + 6m)
(C) the angular speed of the system (bar + particle) after the collision
2 mv
(M + 2m)
A
/2
/2
12 mv
(D) the angular speed of the system (bar + particle) after the collision
(M + 6m)
B
47. A uniform rod of mass 4 m, length 2 (= 2meter) is kept on a smooth horizontal surface. Two balls of
masses m and 3 m moving with velocities u1 = 20 m s−1, u2 = 10 m s−1 respectively, hit simultaneously on
the rod. The direction of their velocities is perpendicular to length of the rod and immediately after
collision both balls come to rest. Then
m u1
2/3
2
C.M.
/3
3m
u2
(A) velocity of the centre of mass of the rod, immediately after collision 12.5 ms−1
(B) angular velocity of the rod about its centre of mass, immediately after collision 2.5 rad s−1
(C) velocity of the centre of mass of the rod, immediately after collision 0
(D) angular velocity of the rod about its centre of mass, immediately after collision 10.5 rad s−1
48. A rod AB of mass M and length L is lying on a horizontal frictionless surface. A particle of mass m
traveling along the surface hits the end ‘A’ of the rod with a velocity v0 in a direction perpendicular to
AB. The collision is completely elastic. The particle comes to rest just after the collision. Then
m 1
(A)
=
M 4
L
(B) a point P which is at rest immediately after collision, its distance from A is
3
v0
(C) the linear speed of the point P of the previous option at a time L/(3v0) after the collision
2 2
v0
(D) the linear speed of the point P of the previous option at a time L/(3v0) after the collision
3 2
7
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49. A disc of radius R is spun to an angular speed o about its axis and then imparted a horizontal velocity of
R
magnitude 0 (at t = 0) with its plane remaining vertical. The coefficient of
4
friction between the disc and the plane is . The sense of rotation and direction
vo
of its linear speed are shown in the figure. Then
o
25
o R
(A) the disc will return to its initial point at t =
48g
(B) during the motion the velocity of CM will become zero before its angular speed becomes zero
25
o R
(C) the disc will return to its initial point at t =
96g
(D) the angular momentum of the disc about a point taken on the ground always remains constant.
50. A solid sphere is set spinning about its centre and is given a translational velocity v when placed on the
horizontal surface as shown in figure. Then
=
2v/R
R
v
v
5
(A) the linear speed of the sphere when it stops rotating
(B) the linear speed of the sphere when slipping finally ceases and pure rolling starts
(C) the linear speed of the sphere when it stops rotating
v
7
(D) the linear speed of the sphere when slipping finally ceases and pure rolling starts
***
8
v
5
v
7