DC Pandey Solutions (JEE Advance): Mechanics of Rotational Motion- 1 | DC Pandey Solutions for JEE Physics PDF Download

 Page 1


For JEE Advanced 
  Assertion and Reason 
  Directions : Choose the correct option. 
  (a) If both Assertion and Reason are true and the Reason is correct explanation of the Assertion. 
  (b) If both Assertion and Reason are true but Reason is not the correct explanation of Assertion. 
  (c) If Assertion is true, but the Reason is false. 
  (d) If Assertion is false but the Reason is true. 
Q 1.  Assertion : Moment of inertia of a rigid body about any axis passing through its centre of mass is 
minimum. 
  Reason : From theorem of parallel axis, 
    I = I cm+Mr
2
 
Q 2.  Assertion : A ball is released on a rough ground in the condition shown in figure. It will start pure 
rolling after some time towards left side. 
 
  Reason: Friction will convert the pure rotational motion of the ball into pure rolling. 
Q 3.  Assertion : A solid sphere and a hollow sphere are rolling on ground with same total kinetic 
energies. If translational kinetic energy of solid sphere is K, then translational kinetic energy of 
hollow sphere should be greater than K. 
Reason : In case of hollow sphere rotational kinetic energy is less than its translational kinetic 
energy. 
Q 4.  Assertion : A small ball is released from rest from point A as shown. If bowl is smooth than ball 
will exert more pressure at point B, compared to the situation if bowl is rough. 
 
  Reason: Linear velocity and hence, centripetal force in smooth situation is more. 
Q 5.  Assertion : A cubical block is moving on a rough ground with velocity v0. During motion net 
normal reaction on the block from ground will not pass through centre of cube. It will shift 
towards right. 
 
  Reason : It is to keep the block in rotational equilibrium. 
Q 6.  Assertion : A ring is rolling without slipping on a rough ground. It strikes elastically with a 
smooth wall as shown in figure. Ring will stop after some time while travelling in opposite 
direction. 
Page 2


For JEE Advanced 
  Assertion and Reason 
  Directions : Choose the correct option. 
  (a) If both Assertion and Reason are true and the Reason is correct explanation of the Assertion. 
  (b) If both Assertion and Reason are true but Reason is not the correct explanation of Assertion. 
  (c) If Assertion is true, but the Reason is false. 
  (d) If Assertion is false but the Reason is true. 
Q 1.  Assertion : Moment of inertia of a rigid body about any axis passing through its centre of mass is 
minimum. 
  Reason : From theorem of parallel axis, 
    I = I cm+Mr
2
 
Q 2.  Assertion : A ball is released on a rough ground in the condition shown in figure. It will start pure 
rolling after some time towards left side. 
 
  Reason: Friction will convert the pure rotational motion of the ball into pure rolling. 
Q 3.  Assertion : A solid sphere and a hollow sphere are rolling on ground with same total kinetic 
energies. If translational kinetic energy of solid sphere is K, then translational kinetic energy of 
hollow sphere should be greater than K. 
Reason : In case of hollow sphere rotational kinetic energy is less than its translational kinetic 
energy. 
Q 4.  Assertion : A small ball is released from rest from point A as shown. If bowl is smooth than ball 
will exert more pressure at point B, compared to the situation if bowl is rough. 
 
  Reason: Linear velocity and hence, centripetal force in smooth situation is more. 
Q 5.  Assertion : A cubical block is moving on a rough ground with velocity v0. During motion net 
normal reaction on the block from ground will not pass through centre of cube. It will shift 
towards right. 
 
  Reason : It is to keep the block in rotational equilibrium. 
Q 6.  Assertion : A ring is rolling without slipping on a rough ground. It strikes elastically with a 
smooth wall as shown in figure. Ring will stop after some time while travelling in opposite 
direction. 
 
Reason : Net angular momentum about an axis passing through bottommost point and 
perpendicular to plane of paper is zero. 
Q 7.  Assertion : There is a thin rod AB and a dotted line CD. All the axes we are talking about are 
perpendicular to plane of paper. As we take different axes moving from A to D, moment of inertia 
of the rod may first decrease then increase. 
 
  Reason : Theorem of perpendicular axis cannot be applied here. 
Q 8.  Assertion : If linear momentum of a particle is constant, then its angular momentum about any 
axis will also remain constant. 
Reason : Linear momentum remain constant, if 
net
F0 ?
 
and angular momentum remains constant 
if 
net
0 ?? . 
Q 9.  Assertion : In the figure shown, A, B and C are three points on the circumference of a disc. Let vA 
, vB and vC are speeds of these three points, then 
 
    vC > v B > v A 
Reason : In case of rotational plus translational motion of a rigid body, net speed of any point 
(other than centre of mass) is greater than, less than or equal to the speed of centre of mass. 
Q 10.  Assertion : There is a triangular plate as shown. A dotted axis is lying in the plane of slab. As the 
axis is moved downwards, moment of inertia of slab will first decrease then increase. 
 
  Reason : Axis is first moving towards its centre of mass and then it is receding from it. 
Q 11.  Assertion : A horizontal force F is applied at the centre of solid sphere placed over a plank. The 
minimum coefficient of friction between plank and sphere required for pure rolling is ?1 when 
plank is kept at rest and ?2 when plank can move, then ?2 < ?1.  
Page 3


For JEE Advanced 
  Assertion and Reason 
  Directions : Choose the correct option. 
  (a) If both Assertion and Reason are true and the Reason is correct explanation of the Assertion. 
  (b) If both Assertion and Reason are true but Reason is not the correct explanation of Assertion. 
  (c) If Assertion is true, but the Reason is false. 
  (d) If Assertion is false but the Reason is true. 
Q 1.  Assertion : Moment of inertia of a rigid body about any axis passing through its centre of mass is 
minimum. 
  Reason : From theorem of parallel axis, 
    I = I cm+Mr
2
 
Q 2.  Assertion : A ball is released on a rough ground in the condition shown in figure. It will start pure 
rolling after some time towards left side. 
 
  Reason: Friction will convert the pure rotational motion of the ball into pure rolling. 
Q 3.  Assertion : A solid sphere and a hollow sphere are rolling on ground with same total kinetic 
energies. If translational kinetic energy of solid sphere is K, then translational kinetic energy of 
hollow sphere should be greater than K. 
Reason : In case of hollow sphere rotational kinetic energy is less than its translational kinetic 
energy. 
Q 4.  Assertion : A small ball is released from rest from point A as shown. If bowl is smooth than ball 
will exert more pressure at point B, compared to the situation if bowl is rough. 
 
  Reason: Linear velocity and hence, centripetal force in smooth situation is more. 
Q 5.  Assertion : A cubical block is moving on a rough ground with velocity v0. During motion net 
normal reaction on the block from ground will not pass through centre of cube. It will shift 
towards right. 
 
  Reason : It is to keep the block in rotational equilibrium. 
Q 6.  Assertion : A ring is rolling without slipping on a rough ground. It strikes elastically with a 
smooth wall as shown in figure. Ring will stop after some time while travelling in opposite 
direction. 
 
Reason : Net angular momentum about an axis passing through bottommost point and 
perpendicular to plane of paper is zero. 
Q 7.  Assertion : There is a thin rod AB and a dotted line CD. All the axes we are talking about are 
perpendicular to plane of paper. As we take different axes moving from A to D, moment of inertia 
of the rod may first decrease then increase. 
 
  Reason : Theorem of perpendicular axis cannot be applied here. 
Q 8.  Assertion : If linear momentum of a particle is constant, then its angular momentum about any 
axis will also remain constant. 
Reason : Linear momentum remain constant, if 
net
F0 ?
 
and angular momentum remains constant 
if 
net
0 ?? . 
Q 9.  Assertion : In the figure shown, A, B and C are three points on the circumference of a disc. Let vA 
, vB and vC are speeds of these three points, then 
 
    vC > v B > v A 
Reason : In case of rotational plus translational motion of a rigid body, net speed of any point 
(other than centre of mass) is greater than, less than or equal to the speed of centre of mass. 
Q 10.  Assertion : There is a triangular plate as shown. A dotted axis is lying in the plane of slab. As the 
axis is moved downwards, moment of inertia of slab will first decrease then increase. 
 
  Reason : Axis is first moving towards its centre of mass and then it is receding from it. 
Q 11.  Assertion : A horizontal force F is applied at the centre of solid sphere placed over a plank. The 
minimum coefficient of friction between plank and sphere required for pure rolling is ?1 when 
plank is kept at rest and ?2 when plank can move, then ?2 < ?1.  
 
  Reason : Work done by frictional force on the sphere in both cases is zero. 
 
Answers 
1. (d) 2. (b) 3. (d) 4. (a) 5. (a) 6. (a) 7. (c) 8. (b) 9. (b) 10. (a) 11. (c) 
 
Solutions 
1.  It depends upon the axis, whether it is parallel or perpendicular to the plane of rotation. So, 
assertion is not always true. 
2.  Due to anti-clockwise rotation, friction will start acting leftward, for which translational speed will 
increase and angular speed will decrease till the moment, when pure rolling starts. So, assertion 
and reason are both true but not complete explanation. 
3.  Hollow sphere has larger moment of inertia, such that it has lesser rotational kinetic energy, so 
assertion is false. 
   RS + KS = RH + KH 
    
    
    
   
4.  Reason is true explanation of assertion. 
5.  Reason is true explanation of assertion. 
6.  Reason is true explanation of assertion. 
7.  As one goes from A to D, distance of CM first decrease and then increase, so moment of inertia 
also first decrease and then increase by applying parallel axes theorem. So, assertion is true but 
reason is false, as perpendicular axis theorem is also applicable. 
8.  If linear momentum is constant, then angular momentum from a particular axis remains constant 
as the projection of distance remains constant. So, assertion and reason are both true but not 
correct explanation. 
Page 4


For JEE Advanced 
  Assertion and Reason 
  Directions : Choose the correct option. 
  (a) If both Assertion and Reason are true and the Reason is correct explanation of the Assertion. 
  (b) If both Assertion and Reason are true but Reason is not the correct explanation of Assertion. 
  (c) If Assertion is true, but the Reason is false. 
  (d) If Assertion is false but the Reason is true. 
Q 1.  Assertion : Moment of inertia of a rigid body about any axis passing through its centre of mass is 
minimum. 
  Reason : From theorem of parallel axis, 
    I = I cm+Mr
2
 
Q 2.  Assertion : A ball is released on a rough ground in the condition shown in figure. It will start pure 
rolling after some time towards left side. 
 
  Reason: Friction will convert the pure rotational motion of the ball into pure rolling. 
Q 3.  Assertion : A solid sphere and a hollow sphere are rolling on ground with same total kinetic 
energies. If translational kinetic energy of solid sphere is K, then translational kinetic energy of 
hollow sphere should be greater than K. 
Reason : In case of hollow sphere rotational kinetic energy is less than its translational kinetic 
energy. 
Q 4.  Assertion : A small ball is released from rest from point A as shown. If bowl is smooth than ball 
will exert more pressure at point B, compared to the situation if bowl is rough. 
 
  Reason: Linear velocity and hence, centripetal force in smooth situation is more. 
Q 5.  Assertion : A cubical block is moving on a rough ground with velocity v0. During motion net 
normal reaction on the block from ground will not pass through centre of cube. It will shift 
towards right. 
 
  Reason : It is to keep the block in rotational equilibrium. 
Q 6.  Assertion : A ring is rolling without slipping on a rough ground. It strikes elastically with a 
smooth wall as shown in figure. Ring will stop after some time while travelling in opposite 
direction. 
 
Reason : Net angular momentum about an axis passing through bottommost point and 
perpendicular to plane of paper is zero. 
Q 7.  Assertion : There is a thin rod AB and a dotted line CD. All the axes we are talking about are 
perpendicular to plane of paper. As we take different axes moving from A to D, moment of inertia 
of the rod may first decrease then increase. 
 
  Reason : Theorem of perpendicular axis cannot be applied here. 
Q 8.  Assertion : If linear momentum of a particle is constant, then its angular momentum about any 
axis will also remain constant. 
Reason : Linear momentum remain constant, if 
net
F0 ?
 
and angular momentum remains constant 
if 
net
0 ?? . 
Q 9.  Assertion : In the figure shown, A, B and C are three points on the circumference of a disc. Let vA 
, vB and vC are speeds of these three points, then 
 
    vC > v B > v A 
Reason : In case of rotational plus translational motion of a rigid body, net speed of any point 
(other than centre of mass) is greater than, less than or equal to the speed of centre of mass. 
Q 10.  Assertion : There is a triangular plate as shown. A dotted axis is lying in the plane of slab. As the 
axis is moved downwards, moment of inertia of slab will first decrease then increase. 
 
  Reason : Axis is first moving towards its centre of mass and then it is receding from it. 
Q 11.  Assertion : A horizontal force F is applied at the centre of solid sphere placed over a plank. The 
minimum coefficient of friction between plank and sphere required for pure rolling is ?1 when 
plank is kept at rest and ?2 when plank can move, then ?2 < ?1.  
 
  Reason : Work done by frictional force on the sphere in both cases is zero. 
 
Answers 
1. (d) 2. (b) 3. (d) 4. (a) 5. (a) 6. (a) 7. (c) 8. (b) 9. (b) 10. (a) 11. (c) 
 
Solutions 
1.  It depends upon the axis, whether it is parallel or perpendicular to the plane of rotation. So, 
assertion is not always true. 
2.  Due to anti-clockwise rotation, friction will start acting leftward, for which translational speed will 
increase and angular speed will decrease till the moment, when pure rolling starts. So, assertion 
and reason are both true but not complete explanation. 
3.  Hollow sphere has larger moment of inertia, such that it has lesser rotational kinetic energy, so 
assertion is false. 
   RS + KS = RH + KH 
    
    
    
   
4.  Reason is true explanation of assertion. 
5.  Reason is true explanation of assertion. 
6.  Reason is true explanation of assertion. 
7.  As one goes from A to D, distance of CM first decrease and then increase, so moment of inertia 
also first decrease and then increase by applying parallel axes theorem. So, assertion is true but 
reason is false, as perpendicular axis theorem is also applicable. 
8.  If linear momentum is constant, then angular momentum from a particular axis remains constant 
as the projection of distance remains constant. So, assertion and reason are both true but not 
correct explanation. 
9.  
  and vA = v - R ? 
  ?  v C > v B > V A 
  So, assertion and reason are both true but not correct explanation. 
10.  Reason is correct explanation of assertion. 
11.  Here, reason is false as when plank was at rest then work done by friction was not zero. 
 
  Objective Questions  
  Single Correct Option 
Q 1.  In the given figure a ring of mass m is kept on a horizontal surface while a body of equal mass m 
is attached through a string, which is wounded on the ring. When the system is released, the ring 
rolls without slipping. Consider the following statement and choose the correct option. 
 
  (i) acceleration of the centre of mass of ring is 
2g
3
  
  (ii) acceleration of hanging particle is 
4g
3
 
  (iii) frictional force (on the ring) acts in forward direction 
  (iv) frictional force (on the ring) acts in backward direction 
  (a) statement (i) and (ii) only are correct  (b) statement (ii) and (iii) only are correct 
  (c) statement (iii) and (iv) only are correct  (d) None of these 
Q 2.  A solid sphere of mass 10 kg is placed on rough surface having coefficient of friction ? = 0.1. A 
constant force F = 7N is applied along a line passing through the centre of the sphere as shown in 
the figure. The value of frictional force on the sphere is 
 
  (a) 1 N   (b) 2 N   (c) 3N    (d) 7N 
Q 3.  From a uniform square plate of side a and mass m, a square portion DEFG of side 
a
2
 is emoved. 
Then, the moment of inertia of remaining portion about the axis AB is 
Page 5


For JEE Advanced 
  Assertion and Reason 
  Directions : Choose the correct option. 
  (a) If both Assertion and Reason are true and the Reason is correct explanation of the Assertion. 
  (b) If both Assertion and Reason are true but Reason is not the correct explanation of Assertion. 
  (c) If Assertion is true, but the Reason is false. 
  (d) If Assertion is false but the Reason is true. 
Q 1.  Assertion : Moment of inertia of a rigid body about any axis passing through its centre of mass is 
minimum. 
  Reason : From theorem of parallel axis, 
    I = I cm+Mr
2
 
Q 2.  Assertion : A ball is released on a rough ground in the condition shown in figure. It will start pure 
rolling after some time towards left side. 
 
  Reason: Friction will convert the pure rotational motion of the ball into pure rolling. 
Q 3.  Assertion : A solid sphere and a hollow sphere are rolling on ground with same total kinetic 
energies. If translational kinetic energy of solid sphere is K, then translational kinetic energy of 
hollow sphere should be greater than K. 
Reason : In case of hollow sphere rotational kinetic energy is less than its translational kinetic 
energy. 
Q 4.  Assertion : A small ball is released from rest from point A as shown. If bowl is smooth than ball 
will exert more pressure at point B, compared to the situation if bowl is rough. 
 
  Reason: Linear velocity and hence, centripetal force in smooth situation is more. 
Q 5.  Assertion : A cubical block is moving on a rough ground with velocity v0. During motion net 
normal reaction on the block from ground will not pass through centre of cube. It will shift 
towards right. 
 
  Reason : It is to keep the block in rotational equilibrium. 
Q 6.  Assertion : A ring is rolling without slipping on a rough ground. It strikes elastically with a 
smooth wall as shown in figure. Ring will stop after some time while travelling in opposite 
direction. 
 
Reason : Net angular momentum about an axis passing through bottommost point and 
perpendicular to plane of paper is zero. 
Q 7.  Assertion : There is a thin rod AB and a dotted line CD. All the axes we are talking about are 
perpendicular to plane of paper. As we take different axes moving from A to D, moment of inertia 
of the rod may first decrease then increase. 
 
  Reason : Theorem of perpendicular axis cannot be applied here. 
Q 8.  Assertion : If linear momentum of a particle is constant, then its angular momentum about any 
axis will also remain constant. 
Reason : Linear momentum remain constant, if 
net
F0 ?
 
and angular momentum remains constant 
if 
net
0 ?? . 
Q 9.  Assertion : In the figure shown, A, B and C are three points on the circumference of a disc. Let vA 
, vB and vC are speeds of these three points, then 
 
    vC > v B > v A 
Reason : In case of rotational plus translational motion of a rigid body, net speed of any point 
(other than centre of mass) is greater than, less than or equal to the speed of centre of mass. 
Q 10.  Assertion : There is a triangular plate as shown. A dotted axis is lying in the plane of slab. As the 
axis is moved downwards, moment of inertia of slab will first decrease then increase. 
 
  Reason : Axis is first moving towards its centre of mass and then it is receding from it. 
Q 11.  Assertion : A horizontal force F is applied at the centre of solid sphere placed over a plank. The 
minimum coefficient of friction between plank and sphere required for pure rolling is ?1 when 
plank is kept at rest and ?2 when plank can move, then ?2 < ?1.  
 
  Reason : Work done by frictional force on the sphere in both cases is zero. 
 
Answers 
1. (d) 2. (b) 3. (d) 4. (a) 5. (a) 6. (a) 7. (c) 8. (b) 9. (b) 10. (a) 11. (c) 
 
Solutions 
1.  It depends upon the axis, whether it is parallel or perpendicular to the plane of rotation. So, 
assertion is not always true. 
2.  Due to anti-clockwise rotation, friction will start acting leftward, for which translational speed will 
increase and angular speed will decrease till the moment, when pure rolling starts. So, assertion 
and reason are both true but not complete explanation. 
3.  Hollow sphere has larger moment of inertia, such that it has lesser rotational kinetic energy, so 
assertion is false. 
   RS + KS = RH + KH 
    
    
    
   
4.  Reason is true explanation of assertion. 
5.  Reason is true explanation of assertion. 
6.  Reason is true explanation of assertion. 
7.  As one goes from A to D, distance of CM first decrease and then increase, so moment of inertia 
also first decrease and then increase by applying parallel axes theorem. So, assertion is true but 
reason is false, as perpendicular axis theorem is also applicable. 
8.  If linear momentum is constant, then angular momentum from a particular axis remains constant 
as the projection of distance remains constant. So, assertion and reason are both true but not 
correct explanation. 
9.  
  and vA = v - R ? 
  ?  v C > v B > V A 
  So, assertion and reason are both true but not correct explanation. 
10.  Reason is correct explanation of assertion. 
11.  Here, reason is false as when plank was at rest then work done by friction was not zero. 
 
  Objective Questions  
  Single Correct Option 
Q 1.  In the given figure a ring of mass m is kept on a horizontal surface while a body of equal mass m 
is attached through a string, which is wounded on the ring. When the system is released, the ring 
rolls without slipping. Consider the following statement and choose the correct option. 
 
  (i) acceleration of the centre of mass of ring is 
2g
3
  
  (ii) acceleration of hanging particle is 
4g
3
 
  (iii) frictional force (on the ring) acts in forward direction 
  (iv) frictional force (on the ring) acts in backward direction 
  (a) statement (i) and (ii) only are correct  (b) statement (ii) and (iii) only are correct 
  (c) statement (iii) and (iv) only are correct  (d) None of these 
Q 2.  A solid sphere of mass 10 kg is placed on rough surface having coefficient of friction ? = 0.1. A 
constant force F = 7N is applied along a line passing through the centre of the sphere as shown in 
the figure. The value of frictional force on the sphere is 
 
  (a) 1 N   (b) 2 N   (c) 3N    (d) 7N 
Q 3.  From a uniform square plate of side a and mass m, a square portion DEFG of side 
a
2
 is emoved. 
Then, the moment of inertia of remaining portion about the axis AB is 
 
 (a) 
2
7ma
16
   (b) 
2
3ma
16
   (c) 
2
3ma
4
   (d) 
2
9ma
16
 
Q 4.  A small solid sphere of mass wand radius r starting from rest from the rim of a fixed 
hemispherical bowl of radius R(>> r) rolls inside it without sliding. The normal reaction exerted 
by the sphere on the hemisphere when it reaches the bottom of hemisphere is 
 
  (a) (3/7) mg   (b) (9/7) mg  (c) (13/7)mg   (d) (17/7)mg 
Q 5.  A uniform solid cylinder of mass m and radius R is placed on a rough horizontal surface. A 
horizontal constant force F is applied at the top point P of the cylinder so that it starts pure rolling. 
The acceleration of the cylinder is 
 
  (a) F/3m   (b) 2F/3m  (c) 4F/3m   (d) 5F/3m 
Q 6.  In the above question, the frictional force on the cylinder is 
  (a) F/3 towards right  (b) F/3 towards left (c) 2F/3 towards right (d) 2F/3 towards left 
Q 7.  A small pulley of radius 20 cm and moment of inertia 0.32kg-m
2
 is used to hang a 2 kg mass with 
the help of massless string. If the block is released, for no slipping condition acceleration of the 
block will be 
 
  (a) 2m/s
2  
(b) 4m/s
2   
(c) 1m/s
2  
(d) 3m/s
2
 
Q 8.  A uniform circular disc of radius R is placed on a smooth horizontal surface with its plane 
horizontal and hinged at circumference through point O as shown. An impulse P is applied at a 
perpendicular distance h from its-centre C. The value of h so that the impulse due to hinge is zero, 
is