IIT JEE- Rotation Motion (MAINS)

Review Paper SCQ 2.

A spool of mass M and internal and external radii R and 2R hanging from a rope touches a curved surface, as shown. A block of mass m placed on a rough surface inclined at an angle  with horizontal is attached with other end of the rope. The pulley is massless and system is just in equilibrium. Find the coefficient of friction. (A) (C)

3mg  2Mg 3mg  2Mg 3mg cos   2Mg sin  3mg sin   2Mg cos 

(B) (D)

3mg sin   2Mg cos  3mg cos   2Mg sin 

3mg  2Mg tan  3mg  2Mg tan  m 

R

2R

2.

B

3.

A disc of radius R is placed on a frictionless x-y plane with origin at centre of disc. At time t = 0 disc is given initial motion such that velocity of point A lying on periphery of disc at x-axis is v0 towards + ve x-axis and angular velocity of disc is   v 0 / 2R. The coordinates

y

A v0

of point A after time t  4R / v 0 are : (A) (4R), 0

(B) (4  1)R, 2R 

(C) (4R), (4R)

(D) None of these.

3.

B

1.

One end of a thin bar AB of mass m and length is hanging from a massless string. The other end is held at an angle of 45° with horizontal. Find the tension in the string immediately after release of the bar. (Bar is released from point B)

x

B

2 (A) mg 5 (C)

1.

A

4 mg 5

3 (B) mg 5 (D)

2 2 mg 5

45° A

3.

Two point masses of 0.3 kg and 0.7 kg are fixed at the ends of a rod of length 1.40 m and of negligible mass. The rod is set rotating about an axis perpendicular to its length with a uniform angular speed. The point on the rod through which the axis should pass in order that the work required for rotation of the rod is minimum, is located at a distance of : (A) 0.98 m from 0.7 kg

(B)

0.98 m from mass of 0.3 kg

(C) 0.70 m from mass of 0.70 kg 3.

B

36.

In the balance machine, shown in figure which arm will move downward ? (A) left (B) right (C) None (D) can not be said

(D) 0.42 m from mass of 0.3 kg

L/2

L/2

3m

m

2m

36.

B.

31.

A constant horizontal force of 5N magnitude is applied tangentially to the top most of a ring of mass 10 kg & radius 10 cm kept on a rough horizontal surface with coeff. of the friction between surface and ring being 0.2. The friction force on the ring would be (g = 10 m/s2)

F(5N) R

(A) 5N directed opposite to the applied force (B) 10 N directed opposite to the applied force (C) 10 N in the direction of applied force (D) none of these 31.

D

7.

A rod of length l falls on two metal pads of same height from a height h. The co-efficients of restitution of the metal pads are e1 and e2 (e1 > e2). The angular velocity of the rod after it recoils is

7.

(A)

e1  2gh e2

(C)

e1  e 2 

C

2gh

(B)

e1  1 2gh e2  1

(D)

e1  1 2gh e2  1

MCQ 8.

A solid uniform sphere is connected with a moving trolley car by a light spring. The trolley car moves with an acceleration a. If the sphere remains at rest relative to the trolley car, then :

a

(A) spring force = ma (B) friction between the sphere and trolley car is equal to zero (C) friction between the sphere and trolley car is equal to (D) spring force is equal to

ma 2

ma 2

8.

A, B

9.

A uniform disc of mass m and radius R starts with velocity V0 on a rough horizontal floor with a purely sliding motion at t = 0. At t = t0, disc starts rolling without sliding. (A) Work done by frictional force upto time t  t 0 is given by

mgt 3gt  2V0  2

(B) Work done by frictional force upto time t  t 0 is given by

mgt  2gt  3V0  2

(C) Work done by frictional force upto time t  2t 0 is given by mgt  3gt  2V0  (D) Work done by frictional force upto time t  2t 0 is given by

mgt 0 3gt0  2V0  2

9.

A, D

10.

Three identical uniform hollow thin pipes are arranged as shown. Pipe and Plank between the pipes have same mass. There is no slipping anywhere. Inclined plane is fixed and has inclination  from horizontal. If system is left free from position shown, choose the correct option (s). (A) Angular acceleration of top pipe is zero (B) acceleration of lowest pipe axis is g sin /2 (C) friction between upper pipe and plank is zero (D) acceleration of plank is g sin 

10. 7.

A, B, C, D A sphere is rotating about a diameter. (A)The particles on the surface of the sphere do not have any linear acceleration. (B)The particles on the diameter mentioned above do not have any linear acceleration (C)Different particles on the surface have different angular speeds. (D)All the particles on the surface have same linear speed.



COMPREHENSION Paragraph for Question Nos. 12 to 13 Read the following write up carefully and answer the following questions: A cylinder of mass m is placed on the edge of a plank of same mass and length 4µgt2 placed on the smooth horizontal surface, where µ is the coefficient of friction between cylinder and plank and t is the time at which pure rolling starts. The cylinder is given an impulse at t = 0 which imparts it a velocity V0 (m/s). 12.

Find the time in which pure rolling starts : (A)

V0 2µg

(B)

V0 4µg

(C)

V0 2 2 µg

(D)

V0 4 2 µg

12.

B

13.

Find the distance covered by cylinder on plank before the start of pure rolling : (A)

13.

gt 2

(B) 2gt 2

(C) 3gt 2

(D) 4gt 2

C

Comprehension – 3 A solid cylinder of mass m and radius R is set in rotation about its axis with an angular velocity , then lowered with its lateral surface onto a horizontal plane and released. The coefficient of friction between the cylinder and the plane is . 21.

The time for which the cylinder will move with sliding is t  (A) 2

22.

(C) 4

(D) 6

The total work performed by friction during the time in which the cylinder move with sliding is   (A) 2

23.

(B) 3

0 R , where x is xg

(B) 3

(C) 4

(D) 6

At the moment when pure rolling starts the velocity of the centre of mass of cylinder is  (A) 2

INTEGER

(B) 3

(C) 4

1 m02R 2 , where x is x

(D) 6

1 0 R , where x is x

1.

A rigid body rotates about a fixed axis. Its angular velocity is variable and is given by ( – t) where  and  are constant and t is the time. The angle through which it rotates before coming to rest is given by  

2 where  is 

an integer. Find . 1.

2

2.

A ball of radius r hits the edge of billiards table with a pure rolling motion and rebounds, with a pure rolling motion. Find the height h (in cm) of the hitting point. Assume that the force exerted on the ball by the edge is horizontal during the impact and that the ball hits the edge along the normal. (Take r = 5 cm).

r

h

2.

7

1.

A solid sphere moving with linear velocity 2m/sec and angular velocity 8 rad/sec is rolling without slipping on a rough horizontal surface to collide elastically with identical sphere at rest of mass 1 kg and radius R. There is no friction between them. Find the ratio of linear velocity of first sphere after it again starts rolling without slipping to the net angular impulse imparted to second sphere by the external forces (an integer).

1.

4

2.

A uniform rod of mass m and length

 3h is in equilibrium. The

value of µ for rod to be in equilibrium is

x . When X is : ( = 60°) 3

µ  h m

2.

1

3.

A uniform rod of mass (m) and length

 5m is released from rest

from its vertical position by giving a gentle push. In consequence, the end of the rod collides at P after rotating about 1 the smooth horizontal axis O. If the coefficient of restitution e  . 2 Angular speed of the rod just after impact is :

µ=0

3 rad/sec where X is X

P

/2 O

3.

2