Rotation and Energy

1. A thin uniform rigid rod AC is shown in Fig (a). It has mass M and
length L = 30cm, is pivoted at point A, and initially held at rest in
a horizontal position. It can rotate about a horizontal axis
perpendicular to the rod passing through A. Another thin,
uniform rigid rod XY of mass M and length l = 75meters is shown
in Fig (b). It is pivoted at point B and initially held at rest in a
horizontal position. It can rotate about a horizontal axis
perpendicular to the rod passing through B. Point B is at a
𝒍
distance of 𝟒 meters from the end X of the rod. Assume g = 10
m/s2.

i. For the rod AC shown in Fig (a), find the maximum angular
velocity it attains when it is released from rest from its
initial horizontal position.
a. 3 𝑟𝑎𝑑 /𝑠 b. 5 𝑟𝑎𝑑 /𝑠
c. 6 𝑟𝑎𝑑 /𝑠 d. 𝟐 𝒓𝒂𝒅 /𝒔

ii. For the rod XY in Fig (b), find the maximum angular
velocity it attains when it is released from rest from its
initial horizontal position. (Take √𝟑 as 1.7)

a.4.8 𝑟𝑎𝑑 /𝑠 b.2.8 𝑟𝑎𝑑 /𝑠

c. 3.8 𝑟𝑎𝑑 /𝑠 d.𝟔. 𝟖 𝒓𝒂𝒅 /𝒔
 Rotation and Energy

2. A uniform ring of mass 5 kg is pivoted at a point A on its

circumference and can rotate in a vertical plane about a horizontal
axis passing through A. The radius of the ring is 2m and initially
it was released from rest with its center at the same horizontal
level as A. Find the maximum angular velocity attained by the
ring. Assume g = 10 m/s2. Give answer upto two places of decimals.

a.𝟐. 𝟐𝟑 𝒓𝒂𝒅 /𝒔 b.2.43 𝑟𝑎𝑑 /𝑠

c.2.25 𝑟𝑎𝑑 /𝑠 d.2.24 𝑟𝑎𝑑 /𝑠
 Rotation and Energy

3. A uniform thin square plate ABCD of mass 6 kg and side length 20

cm is pivoted at point A and can freely rotate about a horizontal
axis passing through A. In the beginning, the square plate was
hanging at rest. It is then hit it with a rod to give it an angular
velocity ω. Find ω, such that the plate just manages to make half
a revolution. Assume g = 10 m/s2.

a.14.55 𝑟𝑎𝑑 /𝑠 b.𝟏𝟒. 𝟓𝟔 𝒓𝒂𝒅 /𝒔

c.14.58 𝑟𝑎𝑑 /𝑠 d.14.52 𝑟𝑎𝑑 /𝑠
 Rotation and Energy

4. Fig (a) shows the initial position of a thin, uniform disc of mass M
𝟓
and radius R = 𝟑 meters that is pivoted at a point A. The point A is
𝑹
at a distance of 𝟐 meters from the center O and vertically below O.
The disc can rotate about a horizontal axis passing through A,
perpendicular to the plane of the disc.
Fig (b) shows the initial position of a thin, uniform disc of mass M
𝒓
and radius r = 2.5meters. Point B is at a distance of 𝟐 meters from
the center O and vertically below O. The disc can rotate about a
horizontal axis in the plane of the disc that passes through B.
Assume g = 10 m/s2.

i. For the disc in Fig (a), find the maximum angular velocity
it attains when it is released from rest.

a.2 𝑟𝑎𝑑 /𝑠 b.6 𝑟𝑎𝑑 /𝑠

c.𝟒 𝒓𝒂𝒅 /𝒔 d.3 𝑟𝑎𝑑 /𝑠

ii. For the disc in Fig (b), find the maximum angular velocity
it attains when it is released from rest.

a.2 𝑟𝑎𝑑 /𝑠 b.6 𝑟𝑎𝑑 /𝑠

c.𝟒 𝒓𝒂𝒅 /𝒔 d.3 𝑟𝑎𝑑 /𝑠
 Rotation and Energy

5. Two thin, uniform rigid rods, each of mass M and length L = 60(√𝟐
− 1) cm, are welded together at the point O as shown in the
figure, so that they form a rigid, right-angled L-shaped object.
The object is pivoted at the point O and initially held at rest in
the position shown, with AO horizontal and OB vertical.
The object can rotate about a horizontal axis passing through O
and perpendicular to the AOB plane. If the L-shaped object is
released from rest, what is the maximum angular velocity that it
attains? Assume g = 10 m/s2.

a.3 𝑟𝑎𝑑 /𝑠 b.𝟓 𝒓𝒂𝒅 /𝒔

c.7 𝑟𝑎𝑑 /𝑠 d.2 𝑟𝑎𝑑 /𝑠
 Rotation and Energy

6. Two thin, uniform rigid rods, each of mass M and length L = 3(√𝟏𝟎
− 1) m, are welded together at the point O as shown in the figure,
to form a rigid right-angled L-shaped object. The object is pivoted
at the point A and initially held at rest in the position shown, with
AO horizontal and OB vertical.
The object can rotate about a horizontal axis passing through A
and perpendicular to the AOB plane. If the L-shaped object is
released from rest, what is the maximum angular velocity that it
attains? Assume g = 10 m/s2.

a.𝟏. 𝟒𝟏𝟒 𝒓𝒂𝒅 /𝒔 b.1.412 𝑟𝑎𝑑 /𝑠

c.1.415 𝑟𝑎𝑑 /𝑠 d.1.419 𝑟𝑎𝑑 /𝑠
 Rotation and Energy

7. A solid uniform disc of mass 10 kg and radius 2 m is pivoted at its

center and can rotate freely about a horizontal axis passing
through the pivot. A massless rope is swung over the disc and from
its two ends, a 7 kg block and a 3 kg block hang as shown. Initially
the blocks are 3 meters above the floor and the system is released
from rest. Assume the rope does not slip on the disc and g = 10
m/s2. Find the speed with which the 7 kg block strikes the ground.

a.2 𝑚/𝑠 b.𝟒 𝒎/𝒔

c.6 𝑚/𝑠 d.8 𝑚/𝑠
 Rotation and Energy

8. At the top of a smooth fixed wedge of incline 300 a uniform solid

disc of mass 4 kg and radius 20 cm is pivoted (at its center). A
massless rope passes over this disc and a 5 kg block hangs
vertically on one side and a 3 kg block placed on the smooth
inclined surface is connected to the other end of the
rope. Initially, the 5 kg block is 1.75 m above the floor and the
system is then released from rest. Assume the rope does not slip
on the disc and g = 10 m/s2. Find the speed with which the 5 kg
block hits the floor.

a.2.5 𝑚/𝑠 b.𝟑. 𝟓 𝒎/𝒔

c.7.5 𝑚/𝑠 d.4.5 𝑚/𝑠
 Rotation and Energy

9. A solid, uniform cylinder of mass M, length L = 600 cm and radius

r = 80π cm, is pivoted horizontally between two fixed walls as
shown in the figure. Two massless ropes are wound (in the same
direction) around the cylinder, and from these ropes two masses M
and 3M hang vertically on the same side of the cylinder. The
system is released from rest. Find the angular velocity of the
cylinder after it completes 9 revolutions. Assume g = 10 m/s2.

a.𝟐𝟎 𝒓𝒂𝒅 /𝒔 b.30 𝑟𝑎𝑑 /𝑠

c.40 𝑟𝑎𝑑 /𝑠 d.50 𝑟𝑎𝑑 /𝑠
 Rotation and Energy

10. Two uniform solid discs, one of mass 3 kg and radius 30 cm,
and the other of mass 1 kg and radius 20 cm are pivoted at their
centres at the ends of a table. They can rotate freely about
horizontal axes through their respective centres. Three blocks of
masses 1 kg, 5 kg and 4 kg are connected together by two light
ropes swung over the two discs as shown.
Assume the ropes do not slip on the discs, the table surface is
frictionless and g = 10m/s^2m/s2. Initially, the 4 kg block is at a
height of H above the ground and then the system is released from
rest. Find H if the 4 kg block strikes the ground with a speed of 2
m/s.

a.60 𝑐𝑚 b.𝟖𝟎 𝒄𝒎
c.20 𝑐𝑚 d.70 𝑐𝑚
 Rotation and Energy

11. Fig (a) shows two uniform, solid pulleys, one of radius R and
mass M, and the other of radius 2R and mass 2M. The pulleys are
fixed at the two ends of a stationary wedge with an angle of
incline 300. A massless rope passes over each of the pulleys and
also connects the blocks M and 5M that hang vertically as shown.
Initially, the block 5M is at a height of 54 cm above the ground.
(Notice that the portion of the rope between the two pulleys is NOT
parallel to the inclined surface of the wedge. But this will not
matter in this question.)
In Fig (b), the same pulleys are fixed at the two ends of the same
wedge. Now, there is a block of mass M on the inclined surface.
There is a massless rope tied to it that passes over the larger pulley
and from the other end of the rope hangs a block of mass 4M.
Another massless rope passes over the smaller pulley and connects
the block M on the incline and a vertically hanging block M.
Initially, the block 4M is at height of 60 cm above the
ground. (Notice that the portions of the ropes connecting the
block M to the two pulleys is parallel to the inclined surface of the
wedge. If it was not, the question would have become significantly
harder.)
Assume the ropes do not slip on the pulleys, the wedge surface is
frictionless and g = 10 m/s2.

i. If the system is released from rest, find the speed with

which the block 5M hits the ground.

a.3.4 𝑚/𝑠 b.𝟐. 𝟒 𝒎/𝒔 c.5.4 𝑚/𝑠 d.7.4 𝑚/𝑠

ii. If the system is released from rest, find the speed with
which the block 4M hits the ground.
a.3 𝑚/𝑠 b.𝟐 𝒎/𝒔 c.7 𝑚/𝑠 d.4 𝑚/𝑠
 Rotation and Energy

12. A uniform solid sphere of radius 30 cm can rotate freely

about a vertical axis passing through its center. A massless string
is wound around the equatorial circle of the sphere and then passes
over a uniform cylinder of radius 17 cm, and from its end hangs a
6 kg block that is initially at a height of 30 cm from the floor. The
cylinder can rotate about its central horizontal axis. Assume g =
10 m/s2 and the string does not slip on the cylinder or the
sphere. The system is releasing from rest. Find the speed of the
block when it hits the floor.

a.2.5 𝑚/𝑠 b.3.5 𝑚/𝑠 c.𝟏. 𝟓 𝒎/𝒔 d.5.5 𝑚/𝑠

 Rotation and Energy
𝟑𝟑𝟎
13. A uniform, solid disc of mass M and radius R = 𝝅+𝟏 cm is
pivoted at its center. A thin, uniform rod of mass M is firmly fixed
on the disc such that it extends from the center of the disc to a
point on its circumference. A massless rope is wound over the disc
and is connected to a block of mass M hanging vertically on one
side, as shown in the figure.

Initially, the system is at rest and the block M is at a height of \pi

RπR meters from the ground. Find the speed with which the block
M hits the ground when the system is released from rest. Assume
the rope does not slip on the disc and g = 10 m/s2.

a.4 𝑚/𝑠 b.2 𝑚/𝑠 c.𝟔 𝒎/𝒔 d.8 𝑚/𝑠

 Rotation and Energy

14. A solid uniform disc of mass 6 kg and radius 2 m is pivoted

at its center. There is a light, circular groove of 1 m radius etched
on the disc and concentric with it. A massless rope is wound over
the disc and a 4 kg block hangs by the rope vertically on one
side. Another massless rope is wound over the groove and a 2 kg
block hangs vertically from it on the same side as shown.

Initially, both the masses hang at the same height. Then the
system is released from rest. Find the height difference between
the two blocks when the angular speed of the disc becomes 2 rad/s.
Assume the ropes do not slip on the disc or the groove and g = 10
m/s2.

a.50 𝑐𝑚 b.𝟔𝟎 𝒄𝒎
c.30 𝑐𝑚 d.40 𝑐𝑚
 Rotation and Energy

15. A uniform, solid disc A of mass 2M and radius R is pivoted at

one end of a wedge that is inclined at an angle of 300 to the
horizontal. At the other end of the incline, another solid disc B of
mass 2M and radius 2R is pivoted. Disc B has a concentric,
massless, circular groove of radius R, etched on it as shown. A
block of mass M is on the inclined surface of the wedge and is
connected by a massless rope that is wound on Disc A and also by
another massless rope that is wound on the groove of Disc B. From
a third massless rope that is wound around the circumference of
Disc B, a block M hangs vertically. Initially the system is at rest
and the hanging block M is at a height of 6 meters above the
ground. Then the system is released from rest. Find the speed with
which the hanging block M hits the ground. Assume that the ropes
do not slip on the discs, the portion of the ropes above the inclined
surface are parallel to it, the inclined surface is frictionless
and g = 10m/s2.

a.4 𝑚/𝑠 b.2 𝑚/𝑠 c.𝟔 𝒎/𝒔 d.8 𝑚/𝑠

 Rotation and Energy

16. A uniform solid disc of mass M and radius R is pivoted at its

center at one end of a wedge that is inclined at an angle of 300 to
the horizontal. At the other end of the incline, a uniform annular
disc of mass M, outer radius 3R and inner radius 2R is pivoted at
its center. A block of mass M is placed on the inclined surface of
the wedge. On one side, a massless rope is tied to it and then wound
around the solid disc. On the other side, another massless rope is
tied to the block and is then wound around the inner
circumference of the annular disc as shown.

The portions of the ropes above the incline are parallel to it.
Another massless rope is wound over the outer circumference of
the annular disc and from its end a block of mass M hangs
vertically. Initially, the system is set into motion by giving the
solid disc an angular speed of 4 rad/s in the anti-clockwise
direction. Then the system is released. Find the maximum height
that the hanging block M can rise up to from its initial position.
Assume that the ropes do not slip on the discs, the incline surface
is frictionless, R = 2 meters and g = 10m/s2.

a.𝟐𝟓. 𝟖 𝒎 b.15.8 𝑚
c.20.8 𝑚 d.22.8 𝑚
 Rotation and Energy

17. In Fig (a), a block of mass M hangs vertically from a thin

massless rope that is wound around the outer circumference of a
uniform annular disc of mass M, having an inner radius R and outer
radius 2R, pivoted at its center. The system is initially held in the
position shown such that block M is at a height of 2 meters above
the ground. Another massless rope wound around the inner
circumference of the annular disc is pulled down vertically with a
constant force F = 7N.
In Fig (b), the same block M hangs vertically from a massless rope
wound around the outer circumference of the same annular disc as
before. Another massless rope wound around the inner
circumference of the annular disc is connected to a massless spring
of spring constant k=33.5N/m. The other end of the spring is fixed
to the ground. The system is initially held at rest in the position
shown such that block M is at a height of 2 meters above the
ground.
Assume that the ropes do not slip on the discs, M = 1 kg
and g=10m/s2.

i. In the system shown in Fig (a), find the speed with which
the block M hits the ground, when the system is released
from rest.
a.𝟒 𝒎/𝒔 b.2 𝑚/𝑠 c.6 𝑚/𝑠 d.8 𝑚/𝑠
ii. In the system shown in Fig (b), find the speed with which
the block M hits the ground, when the system is released
from rest.
a.4 𝑚/𝑠 b.𝟐 𝒎/𝒔 c.6 𝑚/𝑠 d.8 𝑚/𝑠
 Rotation and Energy

18. Fig (a) shows a uniform solid disc of mass M = 2 kg and

radius R = 0.5 meters, pivoted at its center, at the top end of a
wedge inclined at 370. A massless rope is wound around the disc
and connected to a block of mass M placed on the inclined
surface. The coefficient of friction between the block and the
inclined surface is μ=0.25.
In Fig (b), a spring is added to the same system, such that it is
connected to the block M on the inclined plane on one end and to
a fixed wall on the other. The spring constant k = 16N/m.
In both cases of Fig(a) and Fig (b), assume that the disc is initially
rotating with an angular velocity of 8 rad/s in the anti-clockwise
direction.

i. For the system shown in Fig (a), find the distance moved
up on the incline by the block before it comes to a stop.
a.𝟏. 𝟓 𝒎 b.2.5 𝑚
c.3.5 𝑚 d.4.5 𝑚

ii. For the system shown in Fig (b), find the distance moved
up on the incline by the block before it comes to a stop.
a.𝟏 𝒎 b.2 𝑚
c.3 𝑚 d.4 𝑚
 Rotation and Energy

19. In the system shown, there are three blocks A, B and C of

equal mass M. Block A is placed on the inclined surface of a wedge,
inclined at an angle of 370. It is connected by a rope that passes
over a uniform, solid disc of mass M and radius R1 to Block B that
is placed on the horizontal surface of the wedge. Another rope tied
to Block B passes over a uniform, solid disc of mass M and radius R2
, from the end of which Block C hangs vertically. Initially the
system is held at rest in the position shown such that Block C is
at a height of 2 meters above the ground. Then the system is
released from rest.
Assume that the ropes are massless, do not slip on the discs and g
= 10 m/s2. The portion of the rope above the incline is parallel to
the inclined surface. The portions of the ropes above the horizontal
surface of the wedge are horizontal.

i. If all surfaces are smooth, find the speed with which Block
C hits the ground.
a.4 𝑚/𝑠 b.𝟐 𝒎/𝒔 c.6 𝑚/𝑠 d.8 𝑚/𝑠
ii. If only the inclined surface is rough with a friction
coefficient μ = 0.25, and all other surfaces are smooth,
find the speed with which Block C hits the ground.
a.4.414 𝑚/𝑠 b.5.414 𝑚/𝑠
c.𝟏. 𝟒𝟏𝟒 𝒎/𝒔 d.7.414 𝑚/𝑠
 Rotation and Energy

20. In Fig (a), there is a uniform annular disc of mass M, with

inner radius 2r and outer radius 3r that is pivoted at its center. A
rope passes over the outer circumference of the disc and connects
two blocks of mass M and 2M, hanging on either side of the disc.
Initially the system is held at rest in the position shown such that
the block 2M is at a height of 6.7 meters above the ground. Then
the system is released from rest.
In Fig (b), there are three blocks A, B and C of equal mass M. Block
A is placed on the smooth inclined surface of a wedge, inclined at
an angle of 37^o37o. It is connected by a rope that passes over a
uniform annular disc of mass M, inner radius R and outer radius
3R to Block B that is placed on the horizontal surface of the wedge.
Another rope tied to Block B passes over a uniform, solid disc of
mass M and radius 2R, from the end of which Block C hangs
vertically. Initially the system is held at rest in the position shown
such that Block C is at a height of 14.6 meters above the ground.
Then the system is released from rest.
Assume that the ropes are massless, do not slip on the discs and g
= 10 m/s2. The portion of the rope above the incline is parallel to
the inclined surface. The portions of the ropes above the horizontal
surface of the wedge are horizontal.

i. In Fig (a), find the speed with which the block 2M hits the
ground.

a.8 𝑚/𝑠 b.𝟔 𝒎/𝒔 c.3 𝑚/𝑠 d.2 𝑚/𝑠

ii. In Fig (b), find the speed with which Block C hits the ground.

a.8 𝑚/𝑠 b.𝟔 𝒎/𝒔 c.3 𝑚/𝑠 d.2 𝑚/𝑠