# Numerical Problems : Rotational Mechanics

### LEVEL – I

1. A ring having mass M, radius R is kept on horizontal surface as shown in the figure. Find the minimum value of co-efficient of friction so that ring will not slip. Also find

(a) the direction and the magnitude friction acting on the ring.

(b) acceleration of ring.

(c) angular acceleration of ring.

[Ans: F/2M opposite to the applied force , F/2MR clockwise sense ]

2. A disc is rotating about one of its diameters with a kinetic energy E. If the mass and the radius of the disc are m and r respectively, find its angular momentum.

[Ans: R√(mE/2)]

3. A solid uniform disk of mass m and radius R is pivoted about a horizontal axis tangential to the rim of disc. A particle of mass m is attached to a point on the rim of disk, diametrically opposite to the pivot. The combination is now released from rest, with the plane of disc initially horizontal. Find the angular velocity when the small particle reaches its lowest point.

[Ans: ω = √(12g/11r)]

4. The flywheel of a gasoline engine is required to give up 300 J of kinetic energy while its angular velocity decreases from 600 rev min-1 to 540 rev. min-1. What is the moment of inertia of the flywheel ?

[Ans: 0.81 kg-m2]

5. A cord, with one end fixed to a horizontal ceiling, is wrapped over a flywheel of radius ‘ r ‘. The wheel is allowed to fall. Find the angular acceleration of the wheel and the tension in the cord.

[Ans: g/2r ,  mg/2 ]

6. A uniform disc of radius r, and mass ‘ M ‘ kg can rotate without friction about a fixed vertical axis passing through its center and perpendicular to its plane. A cord is wound at the rim of the disc and a uniform force of F Newton is applied on the cord. Find the tangential acceleration of a point on the rim of the disc.

[Ans: 2F/M]

7. A ball is thrown in such a way that it slides with a speed vo initially without rolling on a rough horizontal plane. Prove that it will roll without sliding when its speed falls to (5/7)vo.

8. A disc of mass m , radius r being wrapped over by a light and inextensible string is pulled by force F at the free end of the string. If it moves on a smooth horizontal surface, find (a) linear (b) angular acceleration of the disc.

[Ans: (a)F/m (b)2F/mR ]

9. Show that a cylinder will slip on an inclined plane of inclination θ if the coefficient of static friction between plane and cylinder is less than (1/3)tanθ .

10. A uniform rod of mass M and length a lies on a smooth horizontal plane. A particle of mass m moving at a speed v perpendicular to the length of the rod strikes it at a distance a/4 from the centre and stops after the collision. Find :
(a) the velocity of the centre of the rod and

(b) the angular velocity of the rod about its centre just after the collision.

[Ans: (a)V = mv/M (b)ω = 3mv/Ma ]

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