Lakshay 2013 Physics Assignment Rotational Motion

Assignment (IIT-JEE 2013 -2 Year Program) Date:

PHYSICS Topic: System of Particles & Rotational Motion 1.

Give the location of the centre of mass of a (i) sphere, (ii) cylinder, (iii) ring, and (iv) cube, each of uniform mass density. Does the centre of mass of a body necessarily lie inside the body. [Ans. No]

2.

A child sits stationary at the end of a long trolley moving uniformly with a speed V on a smooth horizontal floor. If the child gets up and runs about on the trolley in any manner, what is the speed of the CM of (trolley + child) system. [Ans. 0]

3.

Show that angular momentum of a satellite of mass ‘Ms’ revolving around the earth having mass Me in an orbit of radius ‘r’ is equal to

4.

GM e M 2s r .

If earth were to shrink suddenly, what would happen to the length of the day. 11

[Ans. decrease]

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5.

The radius of earth’s orbit is 1.5 × 10 m & its period of revolution is 3.2 × 10 s. What will be (a) the magnitude of the velocity (b) the centripetal acceleration of earth in its motion around sun. [Ans. 2.945×104 m/s, 5.78×10–3 m/s2]

6.

A metre scale AB is held vertically with its one end A on the floor & is then allowed to fall. Find the speed of the other end B when its strikes the floor, assuming that the end on the floor doesn’t slip. [Ans. 5.4 m/s]

7.

What will be the duration of day, if the earth suddenly shrinks to 1/64 of its original volume, mass remaining unchanged?

8.

Two bodies of masses 1 kg and 2 kg are located at (1, 2) and (–1, 3) respectively. Calculate the co-ordinates of the centre of mass. 1 8 [Ans.  , ] 3 3

9.

A person is standing on a rotating table with metal spheres in his hands. If he with draws his hands to his chest? What will be the effect on his angular velocity?

10.

Two discs of moment of inertia I1 and I2 about their respective axes (normal to the disc and passing through the centre) and rotating with angular speeds ω1 and ω2 are brought into contact face to face with their axes of rotation coincident. I11  I 22 ] I1  I 2 (b) Show that the K.E. of the combined system is less than the sum of the initial kinetic energies of the two discs. How do you [Ans.  

(a) What is the angular speed of the two-disc system.

account for the loss in energy. Take ω1  ω2 . 11.

An energy of 484 joules is spent in increasing the speed of a flywheel from 60 rpm to 360 rpm. Find the moment of inertia of the flywheel.

12.

Establish a relation between angular momentum and moment of inertia of a rigid body. Define moment of inertia in terms of it. [Ans. 0.7 kgm2]

13.

Obtain an expression for position vector of centre of mass of a two particle system.

14.

A solid sphere rolls down an inclined plane without slipping. Prove that its acceleration down the inclined plane is given by 5 g sin θ . 7

15.

Define a rigid body. Name two kinds of motion which a rigid body can execute. What is the meant by the term equilibrium. For the equilibrium of a body, two conditions need to be satisfied. State them.

16.

Torques of equal magnitude are applied to a hollow cylinder and a solid sphere, both having the same mass and radius. The cylinder is free to rotate about its standard axis of symmetry, and the sphere is free to rotate about an axis passing through its centre. Which of the two will acquire a greater angular speed after a given time. [Ans. sphere]

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Assignment (IIT-JEE 2013 -2 Year Program) Date:

PHYSICS Topic: System of Particles & Rotational Motion 17.

A disc rotating about its axis with angular speed ω0 is place lightly (without any translational push) on a perfectly frictionless table. the radius of the disc is R. What are the linear velocites of the points A, B and C on the disc shown in figure. Will the disc roll in the 3 [Ans. VA  20 R, VB  0, VC  0 R ] 2

direction indicated.

18.

Explain why friction is necessary to make the disc in figure roll in the direction indicated.

(a) Give the direction of frictional force at B, and the sense of frictional torque, before perfect rolling begins. (b) What is the force of friction after perfect rolling begins. [Ans. (a) in backward direction (b) zero] 19.

Read each statement below carefully, and state, with reasions, if it is true or false; (a) During rolling, the force of friction acts in the same direction as the direction of motion of the CM of the body. (b) The instantaneous speed of the point of contact during rolling is zero. (c) For perfect rolling motion, work done agains friction is zero. (d) A wheel moving down a perfectly frictionless inclined plane will undergo slipping (not rolling) motion. [Ans. (a) F (b) T (c) T (d) T]

20.

A cylinder of length 20 cm and radius 10 cm is rotating about its central axis at an angular speed of 100 rad/s. what tangential force will stop the cylinder at a uniform rate in 10 seconds. The moment of inertia of the cylinder about its axis of rotation is 0.8 kg m2. [Ans. 80 N]

21.

Prove that the rate of change of the angular momentum of particle is equal to the torque acting on it.

22.

Define moment of inertia. Write any two factors on which it depends. When the diver leaves the diving board, why does he bring his hand and feet closer together in order to make a somersault.

23.

Three point masses of 1 kg, 2 kg and 3 kg lie at (1, 2), (0 ,–1) and (2, –3) respectively. Calculate the co-ordinates of centre of mass of the system. [Ans. 7/6, –3/2]

24.

A uniform ring of mass 10 kg and diameter 0.40 m rotates with a uniform speed of 2100 rotations per minute. Find the moment of inertia and angular momentum of the ring abut its geometric axis. [Ans. 28 J/kg . m2/s]

25.

A rope of negligible mass is wound round a hollow cylinder of mass 3 kg and radius 40 cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of 30 N. What is the linear acceleration of the rope. [Ans. 10 m/s2]

26.

Three point masses m1, m2 and m3 are located at the vertices of an equilateral triangle of side length a. What is the moment of inertia about an axis along the altitude of the triangle passing through m1.

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[Ans.

1  m1  m2  a 2 ] 4

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Assignment (IIT-JEE 2013 -2 Year Program) Date:

PHYSICS Topic: System of Particles & Rotational Motion 27.

To maintain a rotor at a uniform angular speed of 200 rad s–1 , an engine needs to transmit a torque of 180 Nm. What is the power required by the engine. [Ans. 36000 watts] [Note: Uniform angular velocity in the absence of friction implies zero torque. In practice, applied torque is needed to counter frictional torque]. Assume that the engine is 100% efficient.

28.

A solid cylinder of mass 20 kg rotates about its axis with angular speed 100 rad s–1. The radius of the cylinder is 0.25 m. What is the K.E. associated with the rotation of the cylinder. What is the magnitude of angular momentum of the cylinder about its axis. [Ans. 3125 J]

29.

A solid disc, a ring, both of radius 10 cm are placed on a horizontal table simultaneously, with initial angular speed equal to the 10π rad s–1. Which of the two will start to roll earlier. The coefficient of kinetic friction µ k  0.2 .

30.

[Ans. ring]

A solid cylinder of mass 20 kg and radius 0.12 m rotating with initial angular speed of 125 rad s–1 is placed lightly (i.e., without any translational push) on a horizontal table with coefficient of kinetic friction µ k  0.15 , between the cylinder and the table. (a) After how long does the cylinder start rolling. (b) What is the initial (i) translational energy, (ii) rotational energy and (iii) total energy of the cylinder. (c) After rolling begins, what is (i) translational energy (ii) rotational energy and (iii) total energy. (d) Where does the difference in energies (total) disappear. [Ans. (a) 5s (b)(i) 0 (ii) 1125 J (iii) 1125 J (c) (i) 562.5 J (ii) 0 (iii) 562.5 J

31.

Show that the child’s new kinetic energy of rotation is more than the initial kinetic energy of rotation is more than the initial kinetic energy of rotation. How do you account for this increase in kinetic energy.

32.

A cylinder of mass 10 kg and radius 15 cm is rolling perfectly on a plane of inclination 30°. The co-efficient of static friction µ s  0.25 . (a) How much is the force friction acting on the cylinder. (b) If the inclination θ of the plane is increased, at what value of θ does the cylinder begin to skid, and not roll perfectly.. 1  3  [Ans. (a) 50/3 N (b)   tan   ] 4

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