# Numerical Problems : Rotational Mechanics

#### LEVEL – II

Q:1. A boy stands on a freely rotating platform with his arms stretched. His rotation speed is 0.25 rev./s. But when he draws them in, his speed is 0.80 rev./s.

Find (a) the ratio of his moment of inertia in the first case to that in the second.

(b) the ratio of K.E. in the first case to that in the second.

Ans: (a)16/5 (b) 5/16

Q:2. A sphere of mass m and radius R rolls without sliding on a horizontal surface. It collides with a light spring of stiffness K with a kinetic energy E. If the surface (AB) under the spring is smooth, find the maximum compression of the spring Ans: $\displaystyle = \sqrt{\frac{10E}{7k}}$

Q:3. A uniform rod of mass m & length l0is rotating with a constant angular speed ω about a vertical axis passing through its point of suspension. Find the moment of inertia of the rod about the axis of rotation if it makes an angle θ to the vertical (axis of rotation) Ans: $\displaystyle = \frac{m l_0^2 sin^\theta}{3}$

Q:4. Where should a spherical shell placed on a smooth horizontal surface (shown in the figure) be hit by a cue that it will roll without sliding ? Ans: $\displaystyle \frac{2}{3}R$

Q:5. In the figure shown two particles m & M are interconnected by an inextensible and light string. M is in equilibrium due to revolution of particle m. Now M is pulled down slowly through a distance l/2. Find the change in angular speed of particle m. Ans: $\displaystyle \Delta \omega = 3 \sqrt{\frac{M g}{m l}}$

Q:6. A solid sphere is projected up along an inclined plane of inclination θ =30° with a speed v = 2m/sec. If it rolls without slipping, find the maximum distance traversed by it (g = 10 m/sec2)

Ans: 0.56 m

Q:7. A bullet of mass m collides inelastically at the periphery of a disc of mass M and radius R, with a speed v. The disc rotates about a fixed horizontal axis. Find the angular velocity of the disc bullet system just after the impact. Ans: $\displaystyle \omega = \frac{v}{R(M/2m + 1)}$

Q:8. Two heavy metallic plates are joined together at 900 to each other. A laminar sheet of mass 30Kg is hinged at the line AB joining the two heavy metallic plates. The hinges are frictionless. The moment of inertia of the laminar sheet about an axis parallel to AB and passing through its centre of mass is 1.2 Kg-m2 Two rubber obstacles P and Q are fixed, one on each metallic plates at a distance 0.5 m from the line AB. This distance is chosen so that the reaction due to the hinges on the laminar sheet is zero during the impact. Initially the laminar sheet hits one of the obstacles with an angular velocity 1 rad/s and turns back. If the impulse on the sheet due to each obstacle is 6N-s,

(a) Find the location of the centre of mass of the laminar sheet from AB.

(b) At what angular velocity does the laminar sheet come back after the first impact?

(c) After how may impacts, does the laminar sheets come to rest.

Ans: (a) 0.1 m (b) 1 rad/s (c) Laminar sheet will never come to rest.

Q:9. A boy rolls a hoop over a horizontal path with a speed of 7.2 km/h. Over what distance can the hoop roll uphill at the expense of its kinetic energy? The slope of the hill is 1 in 10.

Ans: 4 m

Q:10. A solid sphere of radius R is moving on a rough horizontal plane. At certain instant, it has translational velocity v0 in right direction and an angular velocity v0/4R in clockwise sense. When its translational velocity is 0.75 v0. It has a perfectly elastic collision with a smooth vertical wall which is normal to its path. Find the speed of the sphere when the sphere starts rolling.

Ans: v0/28

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