Practice Test - Rotational Motion & Gravitational Motion Class 11 Notes | EduRev

Class 11 : Practice Test - Rotational Motion & Gravitational Motion Class 11 Notes | EduRev

 Page 1


                                   [1] 
 
 
 
Test 05 (Rotational Motion & Gravitational Motion) 
1. A thin uniform circular ring is rolling down an 
inclined plane of inclination 30 ° without slipping. 
Its linear acceleration along the inclined plane will 
be :  
(a) g/2  (b) g/3  
(c) g/4 (d) 2 g/3 
2. Which of the following statements about angular 
momentum is correct :  
(a) It is directly proportional to moment of 
inertia  
(b) It is a scalar quantity   
(c) All of these   
(d) None of these  
3. A man stands at one end of a boat which is 
stationary in water. Neglect water resistance. The 
man now moves to the other end of the boat and 
again becomes stationary. The centre of mass of 
the man plus boat system will remain stationary 
with respect to water :  
(a) in all cases   
(b) only when the man is stationary initially and 
finally   
(c) only if the man moves without acceleration 
on the boat   
(d) only if the man and the boat have equal 
masses  
4. What is the moment of inertia of a solid sphere of 
density ? and radius R about its diameter :  
(a) 
176
105
R
5
 ? (b)  
176
105
R
2
? 
(c) 
105
176
R
5
? (d) 
105
176
R
2
? 
5. A stick of length L and mass M lies on a 
frictionless horizontal surface on which it is free to 
move in any way. A ball of mass m moving with 
speed v collides elastically with the stick as shown 
in fig. If after the collision ball comes to rest, then 
what should be the mass of the ball :  
 
(a) m = 2 M  (b) m = M   
(c) m = M/2  (d) m = M/4  
6. A uniform rod of mass m and length ?makes a 
constant angle ? with an axis of rotation which 
passes through one end of the rod. Its moment of 
inertia about this axis is :  
(a) 
3
m
2
?
 (b)  
3
m
2
?
sin ?  
(c) 
3
m
2
?
sin
2
 ?  (d) ?
2
2
cos
3
m ?
 
7. The angular velocity of a body changes from ?
1
 to 
?
2
 without applying torque but by changing 
moment of inertia. The ratio of initial radius of 
gyration to the final radius of gyration is :   
(a) ?
2
 : ?
1
 (b) 
2
1
2
2
: ? ?  
(c) 
1 2
: ? ? (d) 1/ ?
2
 : 1/ ?
1
 
8. Two particles A and B initially at rest, moves 
toward each other under a mutual force of 
attraction. At the instant when the speed of A is v 
and the speed of B is 2v, the speed of centre of 
mass is :  
(a) zero  (b) v   
(c) 1.5 v  (d) 3 v  
9. In rotatory motion, linear velocities of all the 
particles of the body are :   
(a) same  (b) different   
(c) zero  (d) cannot say 
10. A cylinder of water, is rotating about its own axis 
with uniform angular velocity ?. The shape of 
free surface of water will be :  
(a) parabola  (b) elliptical   
(c) circular  (d) spherical  
11. A body rolls down an inclined plane, if the K.E. of 
rotational motion is 40% of its K.E. of translation, 
then the body is :  
(a) cylinder  (b) ring   
(c) solid disc  (d) solid sphere  
12. The moment of inertia of two bodies are I 1
 and I
2
. 
Their geometrical shapes are same, the first made 
of iron and the second aluminum, then :  
(a) I
1
 < I
2 
(b) I
1
 = I
2
  
(c) I
1
 > I
2
  
(d) relation between I
1
 and I
2
 depends on the 
actual shape of the bodies  
13. Radius of a ring is 4 cm and its mass is 10 gm. Its 
moment of inertia about an axis passing through 
its centre and perpendicular to its plane is :  
(a) 160 g - cm
2
 (b) 80 g - cm
2
  
(c) 16 g - cm
2
 (d) None of these 
  
14. A ring and a sphere of the same mass and radius 
are rolling. Their kinetic energies are equal. The 
ratio of their velocities is :   
v 
Page 2


                                   [1] 
 
 
 
Test 05 (Rotational Motion & Gravitational Motion) 
1. A thin uniform circular ring is rolling down an 
inclined plane of inclination 30 ° without slipping. 
Its linear acceleration along the inclined plane will 
be :  
(a) g/2  (b) g/3  
(c) g/4 (d) 2 g/3 
2. Which of the following statements about angular 
momentum is correct :  
(a) It is directly proportional to moment of 
inertia  
(b) It is a scalar quantity   
(c) All of these   
(d) None of these  
3. A man stands at one end of a boat which is 
stationary in water. Neglect water resistance. The 
man now moves to the other end of the boat and 
again becomes stationary. The centre of mass of 
the man plus boat system will remain stationary 
with respect to water :  
(a) in all cases   
(b) only when the man is stationary initially and 
finally   
(c) only if the man moves without acceleration 
on the boat   
(d) only if the man and the boat have equal 
masses  
4. What is the moment of inertia of a solid sphere of 
density ? and radius R about its diameter :  
(a) 
176
105
R
5
 ? (b)  
176
105
R
2
? 
(c) 
105
176
R
5
? (d) 
105
176
R
2
? 
5. A stick of length L and mass M lies on a 
frictionless horizontal surface on which it is free to 
move in any way. A ball of mass m moving with 
speed v collides elastically with the stick as shown 
in fig. If after the collision ball comes to rest, then 
what should be the mass of the ball :  
 
(a) m = 2 M  (b) m = M   
(c) m = M/2  (d) m = M/4  
6. A uniform rod of mass m and length ?makes a 
constant angle ? with an axis of rotation which 
passes through one end of the rod. Its moment of 
inertia about this axis is :  
(a) 
3
m
2
?
 (b)  
3
m
2
?
sin ?  
(c) 
3
m
2
?
sin
2
 ?  (d) ?
2
2
cos
3
m ?
 
7. The angular velocity of a body changes from ?
1
 to 
?
2
 without applying torque but by changing 
moment of inertia. The ratio of initial radius of 
gyration to the final radius of gyration is :   
(a) ?
2
 : ?
1
 (b) 
2
1
2
2
: ? ?  
(c) 
1 2
: ? ? (d) 1/ ?
2
 : 1/ ?
1
 
8. Two particles A and B initially at rest, moves 
toward each other under a mutual force of 
attraction. At the instant when the speed of A is v 
and the speed of B is 2v, the speed of centre of 
mass is :  
(a) zero  (b) v   
(c) 1.5 v  (d) 3 v  
9. In rotatory motion, linear velocities of all the 
particles of the body are :   
(a) same  (b) different   
(c) zero  (d) cannot say 
10. A cylinder of water, is rotating about its own axis 
with uniform angular velocity ?. The shape of 
free surface of water will be :  
(a) parabola  (b) elliptical   
(c) circular  (d) spherical  
11. A body rolls down an inclined plane, if the K.E. of 
rotational motion is 40% of its K.E. of translation, 
then the body is :  
(a) cylinder  (b) ring   
(c) solid disc  (d) solid sphere  
12. The moment of inertia of two bodies are I 1
 and I
2
. 
Their geometrical shapes are same, the first made 
of iron and the second aluminum, then :  
(a) I
1
 < I
2 
(b) I
1
 = I
2
  
(c) I
1
 > I
2
  
(d) relation between I
1
 and I
2
 depends on the 
actual shape of the bodies  
13. Radius of a ring is 4 cm and its mass is 10 gm. Its 
moment of inertia about an axis passing through 
its centre and perpendicular to its plane is :  
(a) 160 g - cm
2
 (b) 80 g - cm
2
  
(c) 16 g - cm
2
 (d) None of these 
  
14. A ring and a sphere of the same mass and radius 
are rolling. Their kinetic energies are equal. The 
ratio of their velocities is :   
v 
                                   [2] 
 
 
 
(a) 10 : 7  (b) 7 : 10   
(c) 2 : 5  (d) 5 : 2  
15. Mass of the earth is 81 times the mass of the moon 
and the distance between the earth and moon is 
60 times the radius of the earth. If R is the radius 
of the earth, then the distance between the moon 
and the point  on the line joining the moon and 
earth where the gravitational force becomes zero 
is :  
(a) 30R  (b) 15R   
(c) 6R  (d) 5R  
16. Two particles of equal mass move in a circle of 
radius r under the action of their mutual 
gravitational attraction. If the mass of each 
particle is M, the speed of each particle is :  
(a) 
r
GM
 (b) 
r 2
GM
 
        (c) 
r 4
GM
  (d) 
r
GM 2
 
17. Suppose the gravitational force varies inversely as 
the nth power of the distance. Then the time 
period of a planet in circular orbit of radius R 
around the sum will be the proportional to :  
(a) R
n
                 (b) R
(n+1)/2
      
(c) R
(n-1)/2
    (d)  R
-n
  
18. The gravitational potential at height h above 
earth’s surface is -5.12 ×10
7
 J/kg and acceleration 
due to gravity at this point is 6.4 ms
-2
. If radius of 
earth is 6400 km, the value of h is :  
(a) 1200 km  (b) 1600 km  
(c) 1800 km    (d) 2400 km  
19. Two bodies with masses M
1
 and M
2
 are initially 
at rest and a distance R apart. They then move 
directly towards one another under the influence 
of their mutual gravitational attraction. What is 
the ratio of the distances traveled by M
1
 to the 
distance traveled by M
2
 ? 
(a)  
2
1
M
M
         (b)  
1
2
M
M
    (c)  1              (d)  
2
1
 
20. The ratio of the kinetic energy required to be 
given to the sattelite to escape earth’s 
gravitational field to the kinetic energy required 
to be given so that the sattelite moves in circular 
orbit just above earth’s atmosphere is: 
(a)  One (b)  Two 
(c)  Half (d)  Infinity 
21. A tunnel is dug along a diameter of the earth of 
mass M e
 and radius R
e
. The force on a particle of 
mass m placed in the tunnel at a distance r from 
the centre is: 
(a)  
3
e
R
m GM e
 r (b)  
r R
m GM
3
e
e
 
(c)  
r
mR GM
3
e e
 (d)  
2
e
e
R
m GM
r 
22. If three particles each of mass M are placed at the 
corners of an equilateral triangle of side a, the 
potential energy of the system and the work done 
of the side of the triangle is changed from a to 2a 
are: 
(a)  
2
a
GM 3
,
a 2
GM 3
 
(b)  -
a
GM 3
2
, 
a 2
GM 3
2
 
(c)  -
2
2
a
GM 3
, 
2
2
a 4
GM 3
   
(d) -
a
GM 3
2
, 
a 2
GM 3
 
23. The orbital speed of Jupiter is: 
(a)  equal to the orbital sped of earth 
(b)  greater than the orbital speed of earth 
(c)  less than the orbital speed of earth 
(d)  proportional to the distance from earth 
24. If the radius of orbit of a satellite is changed by a 
factor of 4, then time period is changed by a factor 
of: 
(a)  4 (b)  6 
(c)  8 (d)  none of these 
25. The ratio of energy required to raise a satellite to a 
height h above the earth’s surface to that required 
to put it into the orbit is: 
(a)  h : R (b)  R : h 
(c)  2h : R (d)  h : 2R 
In every question a statement of ASSERTION is 
followed by a statement of REASON. Mark the 
correct answer out of the following choices:  
If both ASSERTION and REASON are true and reason 
is the correct explanation of the assertion. 
(a) If both ASSERTION and REASON are true 
but reason is not the correct explanation of the 
assertion. 
(b) If ASSERTION is true but REASON is false. 
(c) If ASSERTION is false but REASON is true. 
26. Assertion: Moment of inertia is to rotational 
motion what mass is to translational motion. 
Reason: Moment of inertia of a body is maximum 
about the axis passing through its centre of 
gravity. 
27. Assertion: The centre of mass may lie outside the 
body. 
Page 3


                                   [1] 
 
 
 
Test 05 (Rotational Motion & Gravitational Motion) 
1. A thin uniform circular ring is rolling down an 
inclined plane of inclination 30 ° without slipping. 
Its linear acceleration along the inclined plane will 
be :  
(a) g/2  (b) g/3  
(c) g/4 (d) 2 g/3 
2. Which of the following statements about angular 
momentum is correct :  
(a) It is directly proportional to moment of 
inertia  
(b) It is a scalar quantity   
(c) All of these   
(d) None of these  
3. A man stands at one end of a boat which is 
stationary in water. Neglect water resistance. The 
man now moves to the other end of the boat and 
again becomes stationary. The centre of mass of 
the man plus boat system will remain stationary 
with respect to water :  
(a) in all cases   
(b) only when the man is stationary initially and 
finally   
(c) only if the man moves without acceleration 
on the boat   
(d) only if the man and the boat have equal 
masses  
4. What is the moment of inertia of a solid sphere of 
density ? and radius R about its diameter :  
(a) 
176
105
R
5
 ? (b)  
176
105
R
2
? 
(c) 
105
176
R
5
? (d) 
105
176
R
2
? 
5. A stick of length L and mass M lies on a 
frictionless horizontal surface on which it is free to 
move in any way. A ball of mass m moving with 
speed v collides elastically with the stick as shown 
in fig. If after the collision ball comes to rest, then 
what should be the mass of the ball :  
 
(a) m = 2 M  (b) m = M   
(c) m = M/2  (d) m = M/4  
6. A uniform rod of mass m and length ?makes a 
constant angle ? with an axis of rotation which 
passes through one end of the rod. Its moment of 
inertia about this axis is :  
(a) 
3
m
2
?
 (b)  
3
m
2
?
sin ?  
(c) 
3
m
2
?
sin
2
 ?  (d) ?
2
2
cos
3
m ?
 
7. The angular velocity of a body changes from ?
1
 to 
?
2
 without applying torque but by changing 
moment of inertia. The ratio of initial radius of 
gyration to the final radius of gyration is :   
(a) ?
2
 : ?
1
 (b) 
2
1
2
2
: ? ?  
(c) 
1 2
: ? ? (d) 1/ ?
2
 : 1/ ?
1
 
8. Two particles A and B initially at rest, moves 
toward each other under a mutual force of 
attraction. At the instant when the speed of A is v 
and the speed of B is 2v, the speed of centre of 
mass is :  
(a) zero  (b) v   
(c) 1.5 v  (d) 3 v  
9. In rotatory motion, linear velocities of all the 
particles of the body are :   
(a) same  (b) different   
(c) zero  (d) cannot say 
10. A cylinder of water, is rotating about its own axis 
with uniform angular velocity ?. The shape of 
free surface of water will be :  
(a) parabola  (b) elliptical   
(c) circular  (d) spherical  
11. A body rolls down an inclined plane, if the K.E. of 
rotational motion is 40% of its K.E. of translation, 
then the body is :  
(a) cylinder  (b) ring   
(c) solid disc  (d) solid sphere  
12. The moment of inertia of two bodies are I 1
 and I
2
. 
Their geometrical shapes are same, the first made 
of iron and the second aluminum, then :  
(a) I
1
 < I
2 
(b) I
1
 = I
2
  
(c) I
1
 > I
2
  
(d) relation between I
1
 and I
2
 depends on the 
actual shape of the bodies  
13. Radius of a ring is 4 cm and its mass is 10 gm. Its 
moment of inertia about an axis passing through 
its centre and perpendicular to its plane is :  
(a) 160 g - cm
2
 (b) 80 g - cm
2
  
(c) 16 g - cm
2
 (d) None of these 
  
14. A ring and a sphere of the same mass and radius 
are rolling. Their kinetic energies are equal. The 
ratio of their velocities is :   
v 
                                   [2] 
 
 
 
(a) 10 : 7  (b) 7 : 10   
(c) 2 : 5  (d) 5 : 2  
15. Mass of the earth is 81 times the mass of the moon 
and the distance between the earth and moon is 
60 times the radius of the earth. If R is the radius 
of the earth, then the distance between the moon 
and the point  on the line joining the moon and 
earth where the gravitational force becomes zero 
is :  
(a) 30R  (b) 15R   
(c) 6R  (d) 5R  
16. Two particles of equal mass move in a circle of 
radius r under the action of their mutual 
gravitational attraction. If the mass of each 
particle is M, the speed of each particle is :  
(a) 
r
GM
 (b) 
r 2
GM
 
        (c) 
r 4
GM
  (d) 
r
GM 2
 
17. Suppose the gravitational force varies inversely as 
the nth power of the distance. Then the time 
period of a planet in circular orbit of radius R 
around the sum will be the proportional to :  
(a) R
n
                 (b) R
(n+1)/2
      
(c) R
(n-1)/2
    (d)  R
-n
  
18. The gravitational potential at height h above 
earth’s surface is -5.12 ×10
7
 J/kg and acceleration 
due to gravity at this point is 6.4 ms
-2
. If radius of 
earth is 6400 km, the value of h is :  
(a) 1200 km  (b) 1600 km  
(c) 1800 km    (d) 2400 km  
19. Two bodies with masses M
1
 and M
2
 are initially 
at rest and a distance R apart. They then move 
directly towards one another under the influence 
of their mutual gravitational attraction. What is 
the ratio of the distances traveled by M
1
 to the 
distance traveled by M
2
 ? 
(a)  
2
1
M
M
         (b)  
1
2
M
M
    (c)  1              (d)  
2
1
 
20. The ratio of the kinetic energy required to be 
given to the sattelite to escape earth’s 
gravitational field to the kinetic energy required 
to be given so that the sattelite moves in circular 
orbit just above earth’s atmosphere is: 
(a)  One (b)  Two 
(c)  Half (d)  Infinity 
21. A tunnel is dug along a diameter of the earth of 
mass M e
 and radius R
e
. The force on a particle of 
mass m placed in the tunnel at a distance r from 
the centre is: 
(a)  
3
e
R
m GM e
 r (b)  
r R
m GM
3
e
e
 
(c)  
r
mR GM
3
e e
 (d)  
2
e
e
R
m GM
r 
22. If three particles each of mass M are placed at the 
corners of an equilateral triangle of side a, the 
potential energy of the system and the work done 
of the side of the triangle is changed from a to 2a 
are: 
(a)  
2
a
GM 3
,
a 2
GM 3
 
(b)  -
a
GM 3
2
, 
a 2
GM 3
2
 
(c)  -
2
2
a
GM 3
, 
2
2
a 4
GM 3
   
(d) -
a
GM 3
2
, 
a 2
GM 3
 
23. The orbital speed of Jupiter is: 
(a)  equal to the orbital sped of earth 
(b)  greater than the orbital speed of earth 
(c)  less than the orbital speed of earth 
(d)  proportional to the distance from earth 
24. If the radius of orbit of a satellite is changed by a 
factor of 4, then time period is changed by a factor 
of: 
(a)  4 (b)  6 
(c)  8 (d)  none of these 
25. The ratio of energy required to raise a satellite to a 
height h above the earth’s surface to that required 
to put it into the orbit is: 
(a)  h : R (b)  R : h 
(c)  2h : R (d)  h : 2R 
In every question a statement of ASSERTION is 
followed by a statement of REASON. Mark the 
correct answer out of the following choices:  
If both ASSERTION and REASON are true and reason 
is the correct explanation of the assertion. 
(a) If both ASSERTION and REASON are true 
but reason is not the correct explanation of the 
assertion. 
(b) If ASSERTION is true but REASON is false. 
(c) If ASSERTION is false but REASON is true. 
26. Assertion: Moment of inertia is to rotational 
motion what mass is to translational motion. 
Reason: Moment of inertia of a body is maximum 
about the axis passing through its centre of 
gravity. 
27. Assertion: The centre of mass may lie outside the 
body. 
                                   [3] 
 
 
 
Reason: The centre of mass depends only on the 
size of the body. 
28. Assertion: The value of radius of gyration of a 
body is constant. 
Reason: Radius of gyration is the root mean 
square of the sum of the distances of the particles 
from the axis of rotation for a body having 
uniform density.  
29. Assertion : The orbits of the planets are elliptical.   
Reason     : In planetary motion, the total angular 
momentum remains constant.  
30. Assertion : Moon does not have an atmosphere.   
Reason     : Moon is smaller as compared to earth. 
 
 
 
 
 
 
 
 
 
Test 03  (Rotational Motion & Gravitational Motion) 
ANSWER KEY 
 
1 
C 
6 
C 
11 
D 
16 
C 
21 
A 
26 C 
2 
A 
7 
C 
12 
C 
17 
B 
22 
B 
27 C 
3 
A 
8 
A 
13 
A 
18 
B 
23 
C 
28 D 
4 
C 
9 
B 
14 
B 
19 
B 
24 
C 
29 A 
5 
D 
10 
A 
15 
C 
20 
B 
25 
C 
30 A 
 
 
SOLUTIONS 
1. a = 
) R / K ( 1
sin g
2 2
+
?
 = 4 / g
1 1
30 sin g
=
+
°
 
2. From L = I ?, we find that angular momentum is directly proportional to the moment of inertia. 
3. There are no external horizontal forces acting on the man plus boat’ system. (The forces exerted by the man 
and the boat on each other are internal forces for the system). Therefore, the centre of mass of the system, 
which is initially at rest, will always be at rest. 
4. For solid sphere, I = 
2
MR
5
2
 = 
2 3
R R
3
4
5
2
?
?
?
?
?
?
? p = ? ×
5
R
7
22
15
8
 =  ?
5
R
105
176
 
5. Applying the law of conservation of momentum mv = MV …(1). By conservation of angular momentum 
mv
?
?
?
?
?
?
2
L
 = ?
?
?
?
?
?
?
?
?
12
ML
2
……(2). As the collision is elastic, we have 
2 2 2
I
2
1
MV
2
1
mv
2
1
? + = ….(3). Substituting the 
values, we get m = 
4
M
 
6. Mass of the element = dx
m
?
?
?
?
?
?
?
. M.I. of the element about the axis = ( )
2
sin X dx
m
?
?
?
?
?
?
?
?
. I = 
?
?
'
0
2 2
dx x sin
m
?
 dx = 
?
2
2
sin
3
m ?
  
7. I
1
?
1
 : I
2
?
2
   or  MK
1
2
1
? : MK
2
2
?
2
  or 
?
?
?
?
?
?
?
?
?
?
1
2
2
1
:
K
K
 
Page 4


                                   [1] 
 
 
 
Test 05 (Rotational Motion & Gravitational Motion) 
1. A thin uniform circular ring is rolling down an 
inclined plane of inclination 30 ° without slipping. 
Its linear acceleration along the inclined plane will 
be :  
(a) g/2  (b) g/3  
(c) g/4 (d) 2 g/3 
2. Which of the following statements about angular 
momentum is correct :  
(a) It is directly proportional to moment of 
inertia  
(b) It is a scalar quantity   
(c) All of these   
(d) None of these  
3. A man stands at one end of a boat which is 
stationary in water. Neglect water resistance. The 
man now moves to the other end of the boat and 
again becomes stationary. The centre of mass of 
the man plus boat system will remain stationary 
with respect to water :  
(a) in all cases   
(b) only when the man is stationary initially and 
finally   
(c) only if the man moves without acceleration 
on the boat   
(d) only if the man and the boat have equal 
masses  
4. What is the moment of inertia of a solid sphere of 
density ? and radius R about its diameter :  
(a) 
176
105
R
5
 ? (b)  
176
105
R
2
? 
(c) 
105
176
R
5
? (d) 
105
176
R
2
? 
5. A stick of length L and mass M lies on a 
frictionless horizontal surface on which it is free to 
move in any way. A ball of mass m moving with 
speed v collides elastically with the stick as shown 
in fig. If after the collision ball comes to rest, then 
what should be the mass of the ball :  
 
(a) m = 2 M  (b) m = M   
(c) m = M/2  (d) m = M/4  
6. A uniform rod of mass m and length ?makes a 
constant angle ? with an axis of rotation which 
passes through one end of the rod. Its moment of 
inertia about this axis is :  
(a) 
3
m
2
?
 (b)  
3
m
2
?
sin ?  
(c) 
3
m
2
?
sin
2
 ?  (d) ?
2
2
cos
3
m ?
 
7. The angular velocity of a body changes from ?
1
 to 
?
2
 without applying torque but by changing 
moment of inertia. The ratio of initial radius of 
gyration to the final radius of gyration is :   
(a) ?
2
 : ?
1
 (b) 
2
1
2
2
: ? ?  
(c) 
1 2
: ? ? (d) 1/ ?
2
 : 1/ ?
1
 
8. Two particles A and B initially at rest, moves 
toward each other under a mutual force of 
attraction. At the instant when the speed of A is v 
and the speed of B is 2v, the speed of centre of 
mass is :  
(a) zero  (b) v   
(c) 1.5 v  (d) 3 v  
9. In rotatory motion, linear velocities of all the 
particles of the body are :   
(a) same  (b) different   
(c) zero  (d) cannot say 
10. A cylinder of water, is rotating about its own axis 
with uniform angular velocity ?. The shape of 
free surface of water will be :  
(a) parabola  (b) elliptical   
(c) circular  (d) spherical  
11. A body rolls down an inclined plane, if the K.E. of 
rotational motion is 40% of its K.E. of translation, 
then the body is :  
(a) cylinder  (b) ring   
(c) solid disc  (d) solid sphere  
12. The moment of inertia of two bodies are I 1
 and I
2
. 
Their geometrical shapes are same, the first made 
of iron and the second aluminum, then :  
(a) I
1
 < I
2 
(b) I
1
 = I
2
  
(c) I
1
 > I
2
  
(d) relation between I
1
 and I
2
 depends on the 
actual shape of the bodies  
13. Radius of a ring is 4 cm and its mass is 10 gm. Its 
moment of inertia about an axis passing through 
its centre and perpendicular to its plane is :  
(a) 160 g - cm
2
 (b) 80 g - cm
2
  
(c) 16 g - cm
2
 (d) None of these 
  
14. A ring and a sphere of the same mass and radius 
are rolling. Their kinetic energies are equal. The 
ratio of their velocities is :   
v 
                                   [2] 
 
 
 
(a) 10 : 7  (b) 7 : 10   
(c) 2 : 5  (d) 5 : 2  
15. Mass of the earth is 81 times the mass of the moon 
and the distance between the earth and moon is 
60 times the radius of the earth. If R is the radius 
of the earth, then the distance between the moon 
and the point  on the line joining the moon and 
earth where the gravitational force becomes zero 
is :  
(a) 30R  (b) 15R   
(c) 6R  (d) 5R  
16. Two particles of equal mass move in a circle of 
radius r under the action of their mutual 
gravitational attraction. If the mass of each 
particle is M, the speed of each particle is :  
(a) 
r
GM
 (b) 
r 2
GM
 
        (c) 
r 4
GM
  (d) 
r
GM 2
 
17. Suppose the gravitational force varies inversely as 
the nth power of the distance. Then the time 
period of a planet in circular orbit of radius R 
around the sum will be the proportional to :  
(a) R
n
                 (b) R
(n+1)/2
      
(c) R
(n-1)/2
    (d)  R
-n
  
18. The gravitational potential at height h above 
earth’s surface is -5.12 ×10
7
 J/kg and acceleration 
due to gravity at this point is 6.4 ms
-2
. If radius of 
earth is 6400 km, the value of h is :  
(a) 1200 km  (b) 1600 km  
(c) 1800 km    (d) 2400 km  
19. Two bodies with masses M
1
 and M
2
 are initially 
at rest and a distance R apart. They then move 
directly towards one another under the influence 
of their mutual gravitational attraction. What is 
the ratio of the distances traveled by M
1
 to the 
distance traveled by M
2
 ? 
(a)  
2
1
M
M
         (b)  
1
2
M
M
    (c)  1              (d)  
2
1
 
20. The ratio of the kinetic energy required to be 
given to the sattelite to escape earth’s 
gravitational field to the kinetic energy required 
to be given so that the sattelite moves in circular 
orbit just above earth’s atmosphere is: 
(a)  One (b)  Two 
(c)  Half (d)  Infinity 
21. A tunnel is dug along a diameter of the earth of 
mass M e
 and radius R
e
. The force on a particle of 
mass m placed in the tunnel at a distance r from 
the centre is: 
(a)  
3
e
R
m GM e
 r (b)  
r R
m GM
3
e
e
 
(c)  
r
mR GM
3
e e
 (d)  
2
e
e
R
m GM
r 
22. If three particles each of mass M are placed at the 
corners of an equilateral triangle of side a, the 
potential energy of the system and the work done 
of the side of the triangle is changed from a to 2a 
are: 
(a)  
2
a
GM 3
,
a 2
GM 3
 
(b)  -
a
GM 3
2
, 
a 2
GM 3
2
 
(c)  -
2
2
a
GM 3
, 
2
2
a 4
GM 3
   
(d) -
a
GM 3
2
, 
a 2
GM 3
 
23. The orbital speed of Jupiter is: 
(a)  equal to the orbital sped of earth 
(b)  greater than the orbital speed of earth 
(c)  less than the orbital speed of earth 
(d)  proportional to the distance from earth 
24. If the radius of orbit of a satellite is changed by a 
factor of 4, then time period is changed by a factor 
of: 
(a)  4 (b)  6 
(c)  8 (d)  none of these 
25. The ratio of energy required to raise a satellite to a 
height h above the earth’s surface to that required 
to put it into the orbit is: 
(a)  h : R (b)  R : h 
(c)  2h : R (d)  h : 2R 
In every question a statement of ASSERTION is 
followed by a statement of REASON. Mark the 
correct answer out of the following choices:  
If both ASSERTION and REASON are true and reason 
is the correct explanation of the assertion. 
(a) If both ASSERTION and REASON are true 
but reason is not the correct explanation of the 
assertion. 
(b) If ASSERTION is true but REASON is false. 
(c) If ASSERTION is false but REASON is true. 
26. Assertion: Moment of inertia is to rotational 
motion what mass is to translational motion. 
Reason: Moment of inertia of a body is maximum 
about the axis passing through its centre of 
gravity. 
27. Assertion: The centre of mass may lie outside the 
body. 
                                   [3] 
 
 
 
Reason: The centre of mass depends only on the 
size of the body. 
28. Assertion: The value of radius of gyration of a 
body is constant. 
Reason: Radius of gyration is the root mean 
square of the sum of the distances of the particles 
from the axis of rotation for a body having 
uniform density.  
29. Assertion : The orbits of the planets are elliptical.   
Reason     : In planetary motion, the total angular 
momentum remains constant.  
30. Assertion : Moon does not have an atmosphere.   
Reason     : Moon is smaller as compared to earth. 
 
 
 
 
 
 
 
 
 
Test 03  (Rotational Motion & Gravitational Motion) 
ANSWER KEY 
 
1 
C 
6 
C 
11 
D 
16 
C 
21 
A 
26 C 
2 
A 
7 
C 
12 
C 
17 
B 
22 
B 
27 C 
3 
A 
8 
A 
13 
A 
18 
B 
23 
C 
28 D 
4 
C 
9 
B 
14 
B 
19 
B 
24 
C 
29 A 
5 
D 
10 
A 
15 
C 
20 
B 
25 
C 
30 A 
 
 
SOLUTIONS 
1. a = 
) R / K ( 1
sin g
2 2
+
?
 = 4 / g
1 1
30 sin g
=
+
°
 
2. From L = I ?, we find that angular momentum is directly proportional to the moment of inertia. 
3. There are no external horizontal forces acting on the man plus boat’ system. (The forces exerted by the man 
and the boat on each other are internal forces for the system). Therefore, the centre of mass of the system, 
which is initially at rest, will always be at rest. 
4. For solid sphere, I = 
2
MR
5
2
 = 
2 3
R R
3
4
5
2
?
?
?
?
?
?
? p = ? ×
5
R
7
22
15
8
 =  ?
5
R
105
176
 
5. Applying the law of conservation of momentum mv = MV …(1). By conservation of angular momentum 
mv
?
?
?
?
?
?
2
L
 = ?
?
?
?
?
?
?
?
?
12
ML
2
……(2). As the collision is elastic, we have 
2 2 2
I
2
1
MV
2
1
mv
2
1
? + = ….(3). Substituting the 
values, we get m = 
4
M
 
6. Mass of the element = dx
m
?
?
?
?
?
?
?
. M.I. of the element about the axis = ( )
2
sin X dx
m
?
?
?
?
?
?
?
?
. I = 
?
?
'
0
2 2
dx x sin
m
?
 dx = 
?
2
2
sin
3
m ?
  
7. I
1
?
1
 : I
2
?
2
   or  MK
1
2
1
? : MK
2
2
?
2
  or 
?
?
?
?
?
?
?
?
?
?
1
2
2
1
:
K
K
 
                                   [4] 
 
 
 
8. Force on particle A, F
A
 = m
A
a
A
 = 
t
m
A
?
…(1), Similarly,  F
B
 = M
B
a
B
  = 
t
2 m
B
? ×
….(2) . Now 
) F F (
t
2 m
t
m
B A
B A
=
? ×
=
?
? So,  m
A
 = 2m
B
 For the centre of mass of the system v = 
B A
B B A A
m m
v m v m
+
+
 
or  v = 0
m m
v 2 m v m 2
B A
B B
=
+
-
 
9. From v = r ?, linear velocities (v) for particles at different distances (r) from the axis of rotation are different. 
10. The shape of free surface of water is parabolic, because of difference in centrifugal force (F = m ?
2
, which is 
proportional to r) 
11. 
5
2
100
40
n translatio of . E . K
rotation of . E . K
= = . Now moment of inertia of sphere = 
5
2
MR
2
 
12. ?
1
 > ?
2
 ? I
1
 > I
2
 
13. I = mr
2
 
14. Total KE = 
2 2
I
2
1
Mv
2
1
? + = 
?
?
?
?
?
?
?
?
+
2
2
2 2
R
v
MK Mv
2
1
KE = 
?
?
?
?
?
?
?
?
+
2
2
2
R
K
1 Mv
2
1
 ? 
10
7
R 5
R 2
1
1
R
R
1
1
v
V
2
2
2
2
2
s
2
d
=
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
+
= 
15. Required distance x = 
1
1
81
R 60
1
m
m
d
1
2
+
=
+
= 6R 
16. The particles will always remain diametrically opposite so that the force on each particle will be directed along 
the radius. Considering the circular motion of one particle. 
2
2 2
) r 2 (
Gm
r
m
=
?
 or ? = 
r 4
Gm
.  
17.
 T = 
?
pR 2
, E= 
1 n
2
R
GMm
m
2
1
-
= ? Or  ? = 
2 / 1
1 n
R
GM 2
?
?
?
?
?
?
-
?T = 
2 / ) 1 n (
1 n
R
GM 2
2
R / GM 2
R 2
+
-
×
p
=
p
?T ? R
(n +1)/2 
18. Gravitational potential V = – 
h R
GM
+
. Also, g '= 
2
) h R (
GM
+
. Now V = 
2
) h R (
GM
+
- (R + h) = – g '(R + h) ? – 5.12 ×10
7
 
= – 6.4 (6400 + h) × 10
3 
Solving we get , h = 1600 km 
19. Because gravitational force is the mutual force, hence position of centre of mass remains unaffected. M
1
 R
1
 = 
M
2
R
2
  Or    
1
2
2
1
M
M
R
R
= 
20. 2
R
GM
R
GM 2
m
2
1
m
2
1
) KE (
) KE (
2
0
2
e
orbit
escape
= =
?
?
= 
21. F = 
2
e
2
3
3
e
e
R
m GM
r
m
r
3
4
R
3
4
GM
= × p ×
p
r 
22. U
A
 = Potential energy of the system = 
a
GMM
a
GMM
a
GMM
- - - = – 
a
GM 3
2
, (II) U
B
 = – 
a 2
GMM
a 2
GMM
a 2
GMM
- - = – 
a 2
GM 3
2
. work done = change in potential energy = U
B
 – U
A
 = 
a 2
GM 3
2
 
Page 5


                                   [1] 
 
 
 
Test 05 (Rotational Motion & Gravitational Motion) 
1. A thin uniform circular ring is rolling down an 
inclined plane of inclination 30 ° without slipping. 
Its linear acceleration along the inclined plane will 
be :  
(a) g/2  (b) g/3  
(c) g/4 (d) 2 g/3 
2. Which of the following statements about angular 
momentum is correct :  
(a) It is directly proportional to moment of 
inertia  
(b) It is a scalar quantity   
(c) All of these   
(d) None of these  
3. A man stands at one end of a boat which is 
stationary in water. Neglect water resistance. The 
man now moves to the other end of the boat and 
again becomes stationary. The centre of mass of 
the man plus boat system will remain stationary 
with respect to water :  
(a) in all cases   
(b) only when the man is stationary initially and 
finally   
(c) only if the man moves without acceleration 
on the boat   
(d) only if the man and the boat have equal 
masses  
4. What is the moment of inertia of a solid sphere of 
density ? and radius R about its diameter :  
(a) 
176
105
R
5
 ? (b)  
176
105
R
2
? 
(c) 
105
176
R
5
? (d) 
105
176
R
2
? 
5. A stick of length L and mass M lies on a 
frictionless horizontal surface on which it is free to 
move in any way. A ball of mass m moving with 
speed v collides elastically with the stick as shown 
in fig. If after the collision ball comes to rest, then 
what should be the mass of the ball :  
 
(a) m = 2 M  (b) m = M   
(c) m = M/2  (d) m = M/4  
6. A uniform rod of mass m and length ?makes a 
constant angle ? with an axis of rotation which 
passes through one end of the rod. Its moment of 
inertia about this axis is :  
(a) 
3
m
2
?
 (b)  
3
m
2
?
sin ?  
(c) 
3
m
2
?
sin
2
 ?  (d) ?
2
2
cos
3
m ?
 
7. The angular velocity of a body changes from ?
1
 to 
?
2
 without applying torque but by changing 
moment of inertia. The ratio of initial radius of 
gyration to the final radius of gyration is :   
(a) ?
2
 : ?
1
 (b) 
2
1
2
2
: ? ?  
(c) 
1 2
: ? ? (d) 1/ ?
2
 : 1/ ?
1
 
8. Two particles A and B initially at rest, moves 
toward each other under a mutual force of 
attraction. At the instant when the speed of A is v 
and the speed of B is 2v, the speed of centre of 
mass is :  
(a) zero  (b) v   
(c) 1.5 v  (d) 3 v  
9. In rotatory motion, linear velocities of all the 
particles of the body are :   
(a) same  (b) different   
(c) zero  (d) cannot say 
10. A cylinder of water, is rotating about its own axis 
with uniform angular velocity ?. The shape of 
free surface of water will be :  
(a) parabola  (b) elliptical   
(c) circular  (d) spherical  
11. A body rolls down an inclined plane, if the K.E. of 
rotational motion is 40% of its K.E. of translation, 
then the body is :  
(a) cylinder  (b) ring   
(c) solid disc  (d) solid sphere  
12. The moment of inertia of two bodies are I 1
 and I
2
. 
Their geometrical shapes are same, the first made 
of iron and the second aluminum, then :  
(a) I
1
 < I
2 
(b) I
1
 = I
2
  
(c) I
1
 > I
2
  
(d) relation between I
1
 and I
2
 depends on the 
actual shape of the bodies  
13. Radius of a ring is 4 cm and its mass is 10 gm. Its 
moment of inertia about an axis passing through 
its centre and perpendicular to its plane is :  
(a) 160 g - cm
2
 (b) 80 g - cm
2
  
(c) 16 g - cm
2
 (d) None of these 
  
14. A ring and a sphere of the same mass and radius 
are rolling. Their kinetic energies are equal. The 
ratio of their velocities is :   
v 
                                   [2] 
 
 
 
(a) 10 : 7  (b) 7 : 10   
(c) 2 : 5  (d) 5 : 2  
15. Mass of the earth is 81 times the mass of the moon 
and the distance between the earth and moon is 
60 times the radius of the earth. If R is the radius 
of the earth, then the distance between the moon 
and the point  on the line joining the moon and 
earth where the gravitational force becomes zero 
is :  
(a) 30R  (b) 15R   
(c) 6R  (d) 5R  
16. Two particles of equal mass move in a circle of 
radius r under the action of their mutual 
gravitational attraction. If the mass of each 
particle is M, the speed of each particle is :  
(a) 
r
GM
 (b) 
r 2
GM
 
        (c) 
r 4
GM
  (d) 
r
GM 2
 
17. Suppose the gravitational force varies inversely as 
the nth power of the distance. Then the time 
period of a planet in circular orbit of radius R 
around the sum will be the proportional to :  
(a) R
n
                 (b) R
(n+1)/2
      
(c) R
(n-1)/2
    (d)  R
-n
  
18. The gravitational potential at height h above 
earth’s surface is -5.12 ×10
7
 J/kg and acceleration 
due to gravity at this point is 6.4 ms
-2
. If radius of 
earth is 6400 km, the value of h is :  
(a) 1200 km  (b) 1600 km  
(c) 1800 km    (d) 2400 km  
19. Two bodies with masses M
1
 and M
2
 are initially 
at rest and a distance R apart. They then move 
directly towards one another under the influence 
of their mutual gravitational attraction. What is 
the ratio of the distances traveled by M
1
 to the 
distance traveled by M
2
 ? 
(a)  
2
1
M
M
         (b)  
1
2
M
M
    (c)  1              (d)  
2
1
 
20. The ratio of the kinetic energy required to be 
given to the sattelite to escape earth’s 
gravitational field to the kinetic energy required 
to be given so that the sattelite moves in circular 
orbit just above earth’s atmosphere is: 
(a)  One (b)  Two 
(c)  Half (d)  Infinity 
21. A tunnel is dug along a diameter of the earth of 
mass M e
 and radius R
e
. The force on a particle of 
mass m placed in the tunnel at a distance r from 
the centre is: 
(a)  
3
e
R
m GM e
 r (b)  
r R
m GM
3
e
e
 
(c)  
r
mR GM
3
e e
 (d)  
2
e
e
R
m GM
r 
22. If three particles each of mass M are placed at the 
corners of an equilateral triangle of side a, the 
potential energy of the system and the work done 
of the side of the triangle is changed from a to 2a 
are: 
(a)  
2
a
GM 3
,
a 2
GM 3
 
(b)  -
a
GM 3
2
, 
a 2
GM 3
2
 
(c)  -
2
2
a
GM 3
, 
2
2
a 4
GM 3
   
(d) -
a
GM 3
2
, 
a 2
GM 3
 
23. The orbital speed of Jupiter is: 
(a)  equal to the orbital sped of earth 
(b)  greater than the orbital speed of earth 
(c)  less than the orbital speed of earth 
(d)  proportional to the distance from earth 
24. If the radius of orbit of a satellite is changed by a 
factor of 4, then time period is changed by a factor 
of: 
(a)  4 (b)  6 
(c)  8 (d)  none of these 
25. The ratio of energy required to raise a satellite to a 
height h above the earth’s surface to that required 
to put it into the orbit is: 
(a)  h : R (b)  R : h 
(c)  2h : R (d)  h : 2R 
In every question a statement of ASSERTION is 
followed by a statement of REASON. Mark the 
correct answer out of the following choices:  
If both ASSERTION and REASON are true and reason 
is the correct explanation of the assertion. 
(a) If both ASSERTION and REASON are true 
but reason is not the correct explanation of the 
assertion. 
(b) If ASSERTION is true but REASON is false. 
(c) If ASSERTION is false but REASON is true. 
26. Assertion: Moment of inertia is to rotational 
motion what mass is to translational motion. 
Reason: Moment of inertia of a body is maximum 
about the axis passing through its centre of 
gravity. 
27. Assertion: The centre of mass may lie outside the 
body. 
                                   [3] 
 
 
 
Reason: The centre of mass depends only on the 
size of the body. 
28. Assertion: The value of radius of gyration of a 
body is constant. 
Reason: Radius of gyration is the root mean 
square of the sum of the distances of the particles 
from the axis of rotation for a body having 
uniform density.  
29. Assertion : The orbits of the planets are elliptical.   
Reason     : In planetary motion, the total angular 
momentum remains constant.  
30. Assertion : Moon does not have an atmosphere.   
Reason     : Moon is smaller as compared to earth. 
 
 
 
 
 
 
 
 
 
Test 03  (Rotational Motion & Gravitational Motion) 
ANSWER KEY 
 
1 
C 
6 
C 
11 
D 
16 
C 
21 
A 
26 C 
2 
A 
7 
C 
12 
C 
17 
B 
22 
B 
27 C 
3 
A 
8 
A 
13 
A 
18 
B 
23 
C 
28 D 
4 
C 
9 
B 
14 
B 
19 
B 
24 
C 
29 A 
5 
D 
10 
A 
15 
C 
20 
B 
25 
C 
30 A 
 
 
SOLUTIONS 
1. a = 
) R / K ( 1
sin g
2 2
+
?
 = 4 / g
1 1
30 sin g
=
+
°
 
2. From L = I ?, we find that angular momentum is directly proportional to the moment of inertia. 
3. There are no external horizontal forces acting on the man plus boat’ system. (The forces exerted by the man 
and the boat on each other are internal forces for the system). Therefore, the centre of mass of the system, 
which is initially at rest, will always be at rest. 
4. For solid sphere, I = 
2
MR
5
2
 = 
2 3
R R
3
4
5
2
?
?
?
?
?
?
? p = ? ×
5
R
7
22
15
8
 =  ?
5
R
105
176
 
5. Applying the law of conservation of momentum mv = MV …(1). By conservation of angular momentum 
mv
?
?
?
?
?
?
2
L
 = ?
?
?
?
?
?
?
?
?
12
ML
2
……(2). As the collision is elastic, we have 
2 2 2
I
2
1
MV
2
1
mv
2
1
? + = ….(3). Substituting the 
values, we get m = 
4
M
 
6. Mass of the element = dx
m
?
?
?
?
?
?
?
. M.I. of the element about the axis = ( )
2
sin X dx
m
?
?
?
?
?
?
?
?
. I = 
?
?
'
0
2 2
dx x sin
m
?
 dx = 
?
2
2
sin
3
m ?
  
7. I
1
?
1
 : I
2
?
2
   or  MK
1
2
1
? : MK
2
2
?
2
  or 
?
?
?
?
?
?
?
?
?
?
1
2
2
1
:
K
K
 
                                   [4] 
 
 
 
8. Force on particle A, F
A
 = m
A
a
A
 = 
t
m
A
?
…(1), Similarly,  F
B
 = M
B
a
B
  = 
t
2 m
B
? ×
….(2) . Now 
) F F (
t
2 m
t
m
B A
B A
=
? ×
=
?
? So,  m
A
 = 2m
B
 For the centre of mass of the system v = 
B A
B B A A
m m
v m v m
+
+
 
or  v = 0
m m
v 2 m v m 2
B A
B B
=
+
-
 
9. From v = r ?, linear velocities (v) for particles at different distances (r) from the axis of rotation are different. 
10. The shape of free surface of water is parabolic, because of difference in centrifugal force (F = m ?
2
, which is 
proportional to r) 
11. 
5
2
100
40
n translatio of . E . K
rotation of . E . K
= = . Now moment of inertia of sphere = 
5
2
MR
2
 
12. ?
1
 > ?
2
 ? I
1
 > I
2
 
13. I = mr
2
 
14. Total KE = 
2 2
I
2
1
Mv
2
1
? + = 
?
?
?
?
?
?
?
?
+
2
2
2 2
R
v
MK Mv
2
1
KE = 
?
?
?
?
?
?
?
?
+
2
2
2
R
K
1 Mv
2
1
 ? 
10
7
R 5
R 2
1
1
R
R
1
1
v
V
2
2
2
2
2
s
2
d
=
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
+
= 
15. Required distance x = 
1
1
81
R 60
1
m
m
d
1
2
+
=
+
= 6R 
16. The particles will always remain diametrically opposite so that the force on each particle will be directed along 
the radius. Considering the circular motion of one particle. 
2
2 2
) r 2 (
Gm
r
m
=
?
 or ? = 
r 4
Gm
.  
17.
 T = 
?
pR 2
, E= 
1 n
2
R
GMm
m
2
1
-
= ? Or  ? = 
2 / 1
1 n
R
GM 2
?
?
?
?
?
?
-
?T = 
2 / ) 1 n (
1 n
R
GM 2
2
R / GM 2
R 2
+
-
×
p
=
p
?T ? R
(n +1)/2 
18. Gravitational potential V = – 
h R
GM
+
. Also, g '= 
2
) h R (
GM
+
. Now V = 
2
) h R (
GM
+
- (R + h) = – g '(R + h) ? – 5.12 ×10
7
 
= – 6.4 (6400 + h) × 10
3 
Solving we get , h = 1600 km 
19. Because gravitational force is the mutual force, hence position of centre of mass remains unaffected. M
1
 R
1
 = 
M
2
R
2
  Or    
1
2
2
1
M
M
R
R
= 
20. 2
R
GM
R
GM 2
m
2
1
m
2
1
) KE (
) KE (
2
0
2
e
orbit
escape
= =
?
?
= 
21. F = 
2
e
2
3
3
e
e
R
m GM
r
m
r
3
4
R
3
4
GM
= × p ×
p
r 
22. U
A
 = Potential energy of the system = 
a
GMM
a
GMM
a
GMM
- - - = – 
a
GM 3
2
, (II) U
B
 = – 
a 2
GMM
a 2
GMM
a 2
GMM
- - = – 
a 2
GM 3
2
. work done = change in potential energy = U
B
 – U
A
 = 
a 2
GM 3
2
 
                                   [5] 
 
 
 
23. orbital velocity ?
0
 = 
r
1
r
GM
? (where r is the distance of planet from the sun). Since distance of Jupiter 
from the sun is more than the distance between the sun and the earth, therefore orbital speed of Jupiter rise 
less than the orbital speed of earth. 
24. T ?R
3/2
  
2 / 3
2
1
2
1
R
R
T
T
?
?
?
?
?
?
?
?
= ? T
2
= T
1
(4)
3/2
 = 8 T
1
 
25. Energy required to raise the satellite to a height h. E
1
 = – GMm 
) h R ( R
GMmh
R
1
h R
1
+
=
?
?
?
?
?
?
-
+
 =  
h R
gmRh
) h R ( R
mh gR
2
+
=
+
 
? E
2
= 
2
m
2
1
? = 
h R
gR
m
2
1
2
+
 ? 
R
h 2
E
E
2
1
= 
26. (c) Assertion is true but reason is false. 
Both mass and moment of inertia oppose change of motion. Due to inertia, a body continues in its state of rest 
or uniform motion unless external forces act on it to change its state. For translational motion, the magnitude 
of the external force required depends upon the mass of the body. On the other hand, in rotational motion, an 
external torque is required to change the rotation about an axis of the body. The magnitude of the torque 
required depends upon the distribution of mass about the axis of rotation. Moment of inertia of a body 
depends upon the mass and the axis of rotation.  
27. (c) Assertion is true but reason is false. 
The centre of mass of a body may lie where there is no mass, as for example, the centre of mass of a ring is at 
its centre. By definition, the position vector 
?
r of the mass is equal to 
?
?
?
i
i
m
i r m
 which depends upon the 
distribution of mass. It depends upon the shape and size of the body and can lie within or outside the body.  
28. (d) Assertion is false but reason is true. 
Radius of gyration K is a conceptual distance defined by MK
2
 = 
2
i
n
1 i
i
r m
?
=
 for n particles and . M m
i
i
=
?
 Only 
when the particles are of same mass i.e. the body has a uniform density will K = .
n
r ...... r r
2
n
2
2
2
1
+ + +
 Since r is 
the distance of the particles from the axis of rotation, the value of K will change according to the axis. The 
radius of gyration of a body is, therefore, not constant. 
29. (a) Both assertion and reason are true but reason is not the correct explanation of the assertion. 
According to Kepler’s laws, orbits of the planets are elliptical with sun at one of the foci. The speed of the 
planet and its angular speed keep on changing. Since no torque acts on the planet, its angular momentum 
remains constant.  
30. (a) Both assertion and reason are true but reason is not the correct explanation of the assertion 
The escape velocity from the surface of moon is only 2.5km/s. The molecules of the atmospheric gases on 
the moon had much larger thermal velocities. Hence, the molecules of gases have escaped from the moon 
leaving it without any atmosphere.  
 
 
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