Mechanics of Rotational Motion: JEE Advance (Part- 2) - Physics, Solution by DC Pandey NEET Notes | EduRev

 Page 1


More than One Correct Options 
Q 1.  A mass m of radius r is rolling horizontally without any slip with a linear speed v. It then rolls up 
to a height given by 
2
3v
4g
 
  (a) the body is identified to be a disc or a solid cylinder 
  (b) the body is a solid sphere 
  (c) moment of inertia of the body about instantaneous axis of rotation is 
2
3
mr
2
 
  (d) moment of inertia of the body about instantaneous axis of rotation is 
2
7
mr
5
 
Q 2.  Four identical rods each of mass m and length lare joined to form a rigid square frame. The frame 
lies in the xy plane, with its centre at the origin and the sides parallel to the x and y axes. Its 
moment of inertia about 
  (a) the x-axis is 
2
2
ml
3
    
(b) the z-axis is 
2
4
ml
3
 
  (c) an axis parallel to the z-axis and passing through a corner is 
2
10
ml
3
 
  (d) one side is 
2
5
ml
3
 
Q 3.  A uniform circular ring rolls without slipping on a horizontal surface. At any instant, its position is 
 as shown in the figure. Then 
 
  (a) section ABC has greater kinetic energy than section ADC 
  (b) section BC has greater kinetic energy than section CD 
  (c) section BC has the same kinetic energy as section DA 
  (d) the sections CD and DA have the same kinetic energy 
Q 4.  A cylinder of radius R is to roll without slipping between two planks as shown in the figure. Then 
 
  (a) angular velocity of the cylinder is 
v
R
counter clockwise 
  (b) angular velocity of the cylinder is 
2v
R
clockwise 
  (c) velocity of centre of mass of the cylinder is v towards left 
Page 2


More than One Correct Options 
Q 1.  A mass m of radius r is rolling horizontally without any slip with a linear speed v. It then rolls up 
to a height given by 
2
3v
4g
 
  (a) the body is identified to be a disc or a solid cylinder 
  (b) the body is a solid sphere 
  (c) moment of inertia of the body about instantaneous axis of rotation is 
2
3
mr
2
 
  (d) moment of inertia of the body about instantaneous axis of rotation is 
2
7
mr
5
 
Q 2.  Four identical rods each of mass m and length lare joined to form a rigid square frame. The frame 
lies in the xy plane, with its centre at the origin and the sides parallel to the x and y axes. Its 
moment of inertia about 
  (a) the x-axis is 
2
2
ml
3
    
(b) the z-axis is 
2
4
ml
3
 
  (c) an axis parallel to the z-axis and passing through a corner is 
2
10
ml
3
 
  (d) one side is 
2
5
ml
3
 
Q 3.  A uniform circular ring rolls without slipping on a horizontal surface. At any instant, its position is 
 as shown in the figure. Then 
 
  (a) section ABC has greater kinetic energy than section ADC 
  (b) section BC has greater kinetic energy than section CD 
  (c) section BC has the same kinetic energy as section DA 
  (d) the sections CD and DA have the same kinetic energy 
Q 4.  A cylinder of radius R is to roll without slipping between two planks as shown in the figure. Then 
 
  (a) angular velocity of the cylinder is 
v
R
counter clockwise 
  (b) angular velocity of the cylinder is 
2v
R
clockwise 
  (c) velocity of centre of mass of the cylinder is v towards left 
  (d) velocity of centre of mass of the cylinder is 2v towards right 
Q 5.  A uniform rod of mass m = 2 kg and length l = 0.5 m is sliding along two mutually perpendicular 
smooth walls with the two ends P and Q having velocities v P = 4 m/s and vQ = 3 m/s as shown. 
Then 
 
  (a) The angular velocity of rod, ? = 10rad/s, counter clockwise 
  (b) The angular velocity of rod, ? = 5.0rad/s, counter clockwise 
  (c) The velocity of centre of mass of rod, vcm = 2.5 m/s 
  (d) The total kinetic energy of rod, 
25
K
3
? joule 
Q 6.  A wheel is rolling without slipping on a horizontal plane with velocity v an acceleration a of 
centre of mass as shown in figure. Acceleration at 
 
  (a) A is vertically upwards   (b) B may be vertically downwards 
  (c) C cannot be horizontal   (d) a point on the rim may be horizontal leftwards 
Q 7.  A uniform rod of length l and mass 2 m rests on a smooth horizontal table. A point mass m 
moving horizontally at right angles to the rod with velocity v collides with one end of the rod and 
sticks it. Then 
  (a) angular velocity of the system after collision is 
v
l
 
  (b) angular velocity of the system after collision is 
v
2l
 
  (c) the loss in kinetic energy of the system as a whole as a result of the collision is 
2
mv
6
 
  (d) the loss in kinetic energy of the system as a whole as a result of the collision is 
2
7mv
24
 
Q 8.  A nonuniform ball of radius R and radius of gyration about geometric centre =R/2, is kept on a 
frictionless surface. The geometric centre coincides with the centre of mass. The ball is struck 
horizontally with a sharp impulse = J. The point of application of the impulse is at a height h 
above the surface. Then 
Page 3


More than One Correct Options 
Q 1.  A mass m of radius r is rolling horizontally without any slip with a linear speed v. It then rolls up 
to a height given by 
2
3v
4g
 
  (a) the body is identified to be a disc or a solid cylinder 
  (b) the body is a solid sphere 
  (c) moment of inertia of the body about instantaneous axis of rotation is 
2
3
mr
2
 
  (d) moment of inertia of the body about instantaneous axis of rotation is 
2
7
mr
5
 
Q 2.  Four identical rods each of mass m and length lare joined to form a rigid square frame. The frame 
lies in the xy plane, with its centre at the origin and the sides parallel to the x and y axes. Its 
moment of inertia about 
  (a) the x-axis is 
2
2
ml
3
    
(b) the z-axis is 
2
4
ml
3
 
  (c) an axis parallel to the z-axis and passing through a corner is 
2
10
ml
3
 
  (d) one side is 
2
5
ml
3
 
Q 3.  A uniform circular ring rolls without slipping on a horizontal surface. At any instant, its position is 
 as shown in the figure. Then 
 
  (a) section ABC has greater kinetic energy than section ADC 
  (b) section BC has greater kinetic energy than section CD 
  (c) section BC has the same kinetic energy as section DA 
  (d) the sections CD and DA have the same kinetic energy 
Q 4.  A cylinder of radius R is to roll without slipping between two planks as shown in the figure. Then 
 
  (a) angular velocity of the cylinder is 
v
R
counter clockwise 
  (b) angular velocity of the cylinder is 
2v
R
clockwise 
  (c) velocity of centre of mass of the cylinder is v towards left 
  (d) velocity of centre of mass of the cylinder is 2v towards right 
Q 5.  A uniform rod of mass m = 2 kg and length l = 0.5 m is sliding along two mutually perpendicular 
smooth walls with the two ends P and Q having velocities v P = 4 m/s and vQ = 3 m/s as shown. 
Then 
 
  (a) The angular velocity of rod, ? = 10rad/s, counter clockwise 
  (b) The angular velocity of rod, ? = 5.0rad/s, counter clockwise 
  (c) The velocity of centre of mass of rod, vcm = 2.5 m/s 
  (d) The total kinetic energy of rod, 
25
K
3
? joule 
Q 6.  A wheel is rolling without slipping on a horizontal plane with velocity v an acceleration a of 
centre of mass as shown in figure. Acceleration at 
 
  (a) A is vertically upwards   (b) B may be vertically downwards 
  (c) C cannot be horizontal   (d) a point on the rim may be horizontal leftwards 
Q 7.  A uniform rod of length l and mass 2 m rests on a smooth horizontal table. A point mass m 
moving horizontally at right angles to the rod with velocity v collides with one end of the rod and 
sticks it. Then 
  (a) angular velocity of the system after collision is 
v
l
 
  (b) angular velocity of the system after collision is 
v
2l
 
  (c) the loss in kinetic energy of the system as a whole as a result of the collision is 
2
mv
6
 
  (d) the loss in kinetic energy of the system as a whole as a result of the collision is 
2
7mv
24
 
Q 8.  A nonuniform ball of radius R and radius of gyration about geometric centre =R/2, is kept on a 
frictionless surface. The geometric centre coincides with the centre of mass. The ball is struck 
horizontally with a sharp impulse = J. The point of application of the impulse is at a height h 
above the surface. Then 
  (a) the ball with slip on surface for all cases  (b) the ball will roll purely if h = 5R/4  
  (c) the ball will roll purely if h = 3R/2  (d) there will be no rotation if h = R 
Q 9.  A hollow spherical ball is given an initial push up an incline of inclination angle ?.
 
The ball rolls 
purely. Coefficient of static friction between ball and incline = ?. During its upwards journey 
  (a) friction acts up along the incline   (b) (2 tan ?)/5 
  (c) friction acts down along the incline  (d) (2 tan ?)/7 
Q 10.  A uniform disc of mass m and radius R rotates about a fixed vertical axis passing through its 
centre with angular velocity ?. A particle of same mass m and having velocity of 2 ?R towards 
centre of the disc collides with the disc moving horizontally and sticks to its rim. 
  (a) The angular velocity of the disc will become ?/3  
  (b) The angular velocity of the disc will become 5 ?/3  
  (c) The impulse on the particle due to disc is 
37
mR
3
? 
  (d) The impulse on the particle due to disc is 2m ?R 
Q 11.  The end B of the rod AB which makes angle ? with the floor is being pulled with a constant 
velocity v0 as shown. The length of the rod is l. 
 
  (a) At ? = 37 ?
 
velocity of end A is 
0
4
v
3
downwards  
  (b) At ? = 37 ? angular velocity of rod is 
0
5v
3l
 
  (c) Angular velocity of rod is constant 
  (d) Velocity of end A is constant 
Answers 
  1. (a,c) 2. (all) 3. (a,b,d) 4. (a,b) 5. (a,c,d) 6. (all) 7. (a,c) 8. (b,d) 9. (a,b) 10. (a,c) 11. (a,b) 
 
Solutions 
1.   
   
Page 4


More than One Correct Options 
Q 1.  A mass m of radius r is rolling horizontally without any slip with a linear speed v. It then rolls up 
to a height given by 
2
3v
4g
 
  (a) the body is identified to be a disc or a solid cylinder 
  (b) the body is a solid sphere 
  (c) moment of inertia of the body about instantaneous axis of rotation is 
2
3
mr
2
 
  (d) moment of inertia of the body about instantaneous axis of rotation is 
2
7
mr
5
 
Q 2.  Four identical rods each of mass m and length lare joined to form a rigid square frame. The frame 
lies in the xy plane, with its centre at the origin and the sides parallel to the x and y axes. Its 
moment of inertia about 
  (a) the x-axis is 
2
2
ml
3
    
(b) the z-axis is 
2
4
ml
3
 
  (c) an axis parallel to the z-axis and passing through a corner is 
2
10
ml
3
 
  (d) one side is 
2
5
ml
3
 
Q 3.  A uniform circular ring rolls without slipping on a horizontal surface. At any instant, its position is 
 as shown in the figure. Then 
 
  (a) section ABC has greater kinetic energy than section ADC 
  (b) section BC has greater kinetic energy than section CD 
  (c) section BC has the same kinetic energy as section DA 
  (d) the sections CD and DA have the same kinetic energy 
Q 4.  A cylinder of radius R is to roll without slipping between two planks as shown in the figure. Then 
 
  (a) angular velocity of the cylinder is 
v
R
counter clockwise 
  (b) angular velocity of the cylinder is 
2v
R
clockwise 
  (c) velocity of centre of mass of the cylinder is v towards left 
  (d) velocity of centre of mass of the cylinder is 2v towards right 
Q 5.  A uniform rod of mass m = 2 kg and length l = 0.5 m is sliding along two mutually perpendicular 
smooth walls with the two ends P and Q having velocities v P = 4 m/s and vQ = 3 m/s as shown. 
Then 
 
  (a) The angular velocity of rod, ? = 10rad/s, counter clockwise 
  (b) The angular velocity of rod, ? = 5.0rad/s, counter clockwise 
  (c) The velocity of centre of mass of rod, vcm = 2.5 m/s 
  (d) The total kinetic energy of rod, 
25
K
3
? joule 
Q 6.  A wheel is rolling without slipping on a horizontal plane with velocity v an acceleration a of 
centre of mass as shown in figure. Acceleration at 
 
  (a) A is vertically upwards   (b) B may be vertically downwards 
  (c) C cannot be horizontal   (d) a point on the rim may be horizontal leftwards 
Q 7.  A uniform rod of length l and mass 2 m rests on a smooth horizontal table. A point mass m 
moving horizontally at right angles to the rod with velocity v collides with one end of the rod and 
sticks it. Then 
  (a) angular velocity of the system after collision is 
v
l
 
  (b) angular velocity of the system after collision is 
v
2l
 
  (c) the loss in kinetic energy of the system as a whole as a result of the collision is 
2
mv
6
 
  (d) the loss in kinetic energy of the system as a whole as a result of the collision is 
2
7mv
24
 
Q 8.  A nonuniform ball of radius R and radius of gyration about geometric centre =R/2, is kept on a 
frictionless surface. The geometric centre coincides with the centre of mass. The ball is struck 
horizontally with a sharp impulse = J. The point of application of the impulse is at a height h 
above the surface. Then 
  (a) the ball with slip on surface for all cases  (b) the ball will roll purely if h = 5R/4  
  (c) the ball will roll purely if h = 3R/2  (d) there will be no rotation if h = R 
Q 9.  A hollow spherical ball is given an initial push up an incline of inclination angle ?.
 
The ball rolls 
purely. Coefficient of static friction between ball and incline = ?. During its upwards journey 
  (a) friction acts up along the incline   (b) (2 tan ?)/5 
  (c) friction acts down along the incline  (d) (2 tan ?)/7 
Q 10.  A uniform disc of mass m and radius R rotates about a fixed vertical axis passing through its 
centre with angular velocity ?. A particle of same mass m and having velocity of 2 ?R towards 
centre of the disc collides with the disc moving horizontally and sticks to its rim. 
  (a) The angular velocity of the disc will become ?/3  
  (b) The angular velocity of the disc will become 5 ?/3  
  (c) The impulse on the particle due to disc is 
37
mR
3
? 
  (d) The impulse on the particle due to disc is 2m ?R 
Q 11.  The end B of the rod AB which makes angle ? with the floor is being pulled with a constant 
velocity v0 as shown. The length of the rod is l. 
 
  (a) At ? = 37 ?
 
velocity of end A is 
0
4
v
3
downwards  
  (b) At ? = 37 ? angular velocity of rod is 
0
5v
3l
 
  (c) Angular velocity of rod is constant 
  (d) Velocity of end A is constant 
Answers 
  1. (a,c) 2. (all) 3. (a,b,d) 4. (a,b) 5. (a,c,d) 6. (all) 7. (a,c) 8. (b,d) 9. (a,b) 10. (a,c) 11. (a,b) 
 
Solutions 
1.   
   
   Disc or cylinder I0 = I CM + mR
2
 
    
2.  
     
   
    
3.  
  Resultant velocity of particles in region ABC is greater than that in CDA.   
   So,(KE)ABC>(KE)CDA 
  Similarly, vBC > v CD ? K BC >K CD 
  and   vCD = vDA ? K CD =KDA 
Page 5


More than One Correct Options 
Q 1.  A mass m of radius r is rolling horizontally without any slip with a linear speed v. It then rolls up 
to a height given by 
2
3v
4g
 
  (a) the body is identified to be a disc or a solid cylinder 
  (b) the body is a solid sphere 
  (c) moment of inertia of the body about instantaneous axis of rotation is 
2
3
mr
2
 
  (d) moment of inertia of the body about instantaneous axis of rotation is 
2
7
mr
5
 
Q 2.  Four identical rods each of mass m and length lare joined to form a rigid square frame. The frame 
lies in the xy plane, with its centre at the origin and the sides parallel to the x and y axes. Its 
moment of inertia about 
  (a) the x-axis is 
2
2
ml
3
    
(b) the z-axis is 
2
4
ml
3
 
  (c) an axis parallel to the z-axis and passing through a corner is 
2
10
ml
3
 
  (d) one side is 
2
5
ml
3
 
Q 3.  A uniform circular ring rolls without slipping on a horizontal surface. At any instant, its position is 
 as shown in the figure. Then 
 
  (a) section ABC has greater kinetic energy than section ADC 
  (b) section BC has greater kinetic energy than section CD 
  (c) section BC has the same kinetic energy as section DA 
  (d) the sections CD and DA have the same kinetic energy 
Q 4.  A cylinder of radius R is to roll without slipping between two planks as shown in the figure. Then 
 
  (a) angular velocity of the cylinder is 
v
R
counter clockwise 
  (b) angular velocity of the cylinder is 
2v
R
clockwise 
  (c) velocity of centre of mass of the cylinder is v towards left 
  (d) velocity of centre of mass of the cylinder is 2v towards right 
Q 5.  A uniform rod of mass m = 2 kg and length l = 0.5 m is sliding along two mutually perpendicular 
smooth walls with the two ends P and Q having velocities v P = 4 m/s and vQ = 3 m/s as shown. 
Then 
 
  (a) The angular velocity of rod, ? = 10rad/s, counter clockwise 
  (b) The angular velocity of rod, ? = 5.0rad/s, counter clockwise 
  (c) The velocity of centre of mass of rod, vcm = 2.5 m/s 
  (d) The total kinetic energy of rod, 
25
K
3
? joule 
Q 6.  A wheel is rolling without slipping on a horizontal plane with velocity v an acceleration a of 
centre of mass as shown in figure. Acceleration at 
 
  (a) A is vertically upwards   (b) B may be vertically downwards 
  (c) C cannot be horizontal   (d) a point on the rim may be horizontal leftwards 
Q 7.  A uniform rod of length l and mass 2 m rests on a smooth horizontal table. A point mass m 
moving horizontally at right angles to the rod with velocity v collides with one end of the rod and 
sticks it. Then 
  (a) angular velocity of the system after collision is 
v
l
 
  (b) angular velocity of the system after collision is 
v
2l
 
  (c) the loss in kinetic energy of the system as a whole as a result of the collision is 
2
mv
6
 
  (d) the loss in kinetic energy of the system as a whole as a result of the collision is 
2
7mv
24
 
Q 8.  A nonuniform ball of radius R and radius of gyration about geometric centre =R/2, is kept on a 
frictionless surface. The geometric centre coincides with the centre of mass. The ball is struck 
horizontally with a sharp impulse = J. The point of application of the impulse is at a height h 
above the surface. Then 
  (a) the ball with slip on surface for all cases  (b) the ball will roll purely if h = 5R/4  
  (c) the ball will roll purely if h = 3R/2  (d) there will be no rotation if h = R 
Q 9.  A hollow spherical ball is given an initial push up an incline of inclination angle ?.
 
The ball rolls 
purely. Coefficient of static friction between ball and incline = ?. During its upwards journey 
  (a) friction acts up along the incline   (b) (2 tan ?)/5 
  (c) friction acts down along the incline  (d) (2 tan ?)/7 
Q 10.  A uniform disc of mass m and radius R rotates about a fixed vertical axis passing through its 
centre with angular velocity ?. A particle of same mass m and having velocity of 2 ?R towards 
centre of the disc collides with the disc moving horizontally and sticks to its rim. 
  (a) The angular velocity of the disc will become ?/3  
  (b) The angular velocity of the disc will become 5 ?/3  
  (c) The impulse on the particle due to disc is 
37
mR
3
? 
  (d) The impulse on the particle due to disc is 2m ?R 
Q 11.  The end B of the rod AB which makes angle ? with the floor is being pulled with a constant 
velocity v0 as shown. The length of the rod is l. 
 
  (a) At ? = 37 ?
 
velocity of end A is 
0
4
v
3
downwards  
  (b) At ? = 37 ? angular velocity of rod is 
0
5v
3l
 
  (c) Angular velocity of rod is constant 
  (d) Velocity of end A is constant 
Answers 
  1. (a,c) 2. (all) 3. (a,b,d) 4. (a,b) 5. (a,c,d) 6. (all) 7. (a,c) 8. (b,d) 9. (a,b) 10. (a,c) 11. (a,b) 
 
Solutions 
1.   
   
   Disc or cylinder I0 = I CM + mR
2
 
    
2.  
     
   
    
3.  
  Resultant velocity of particles in region ABC is greater than that in CDA.   
   So,(KE)ABC>(KE)CDA 
  Similarly, vBC > v CD ? K BC >K CD 
  and   vCD = vDA ? K CD =KDA 
4.  
   u - R ? = v 
   u + R ? = 3v 
  ? u=2v and R ? = v 
5.  
    
    
6.  
Net acceleration at A is vertical, at B it may be vertical, at C it cannot be horizontal and at E it 
may be horizontal leftward. 
7.