A car is moving with a speed of 30 m/s on a circular path of radius 500 m. If its speed is increasing at the rate of 2 m/s, the net acceleration of the car is

A stone of mass 0.5 kg is attached to a string of length 2 m and is whirled in a horizontal circle. If the string can with stand a tension of 9N, the maximum velocity with which the stone can be whirled is

A car weighs 1800 kg. The distance between its front and back axles is 1.8 m. Its centre of gravity is 1.05 m behind the front axle. Determine the force exerted by the level ground on each front wheel and each back wheel.

In the HCl molecule, the separation between the nuclei of the two atoms is about $1.27 \AA$ ($1 \AA = 10^{-10} m$). Find the approximate location of the CM of the molecule, given that a chlorine atom is about $35.5$ times as massive as a hydrogen atom and nearly all the mass of an atom is concentrated in its nucleus.

Explain why friction is necessary to make the disc in Fig. 7.41 roll in the direction indicated. (b) What is the force of friction after perfect rolling begins ?

A man stands on a rotating platform, with his arms stretched horizontally holding a 5 kg weight in each hand. The angular speed of the platform is 30 revolutions per minute. The man then brings his arms close to his body with the distance of each weight from the axis changing from 90cm to 20cm. The moment of inertia of the man together with the platform may be taken to be constant and equal to $7.6 kg m^2$. (a) What is his new angular speed? (Neglect friction.)

A child sits stationary at one 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 the (trolley + child) system ?

Explain why friction is necessary to make the disc in Fig. 7.41 roll in the direction indicated. (a) Give the direction of frictional force at B, and the sense of frictional torque, before perfect rolling begins.

Read each statement below carefully, and state, with reasons, if it is true or false; (d) For perfect rolling motion, work done against friction is zero.

A solid cylinder rolls up an inclined plane of angle of inclination $30^\circ$. At the bottom of the inclined plane the centre of mass of the cylinder has a speed of 5 m/s. (b) How long will it take to return to the bottom?

A non-uniform bar of weight $W$ is suspended at rest by two strings of negligible weight as shown in Fig.7.39. The angles made by the strings with the vertical are $36.9^\circ$ and $53.1^\circ$ respectively. The bar is 2 m long. Calculate the distance $d$ of the centre of gravity of the bar from its left end.

A cylinder of mass 10 kg and radius 15 cm is rolling perfectly on a plane of inclination $30^o$. The co-efficient of static friction $\mu_s = 0.25$. (a) How much is the force of friction acting on the cylinder ?

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 ? Assume that there is no slipping.

Give the location of the centre of mass of a (iv) cube, each of uniform mass density. Does the centre of mass of a body necessarily lie inside the body ?

A solid disc and a ring, both of radius 10 cm are placed on a horizontal table simultaneously, with initial angular speed equal to $10 \pi$ rad $s^{-1}$. Which of the two will start to roll earlier ? The co-efficient of kinetic friction is $\mu_k = 0.2$.

Read each statement below carefully, and state, with reasons, if it is true or false; (c) The instantaneous acceleration of the point of contact during rolling is zero.

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

As shown in Fig.7.40, the two sides of a step ladder BA and CA are 1.6 m long and hinged at A. A rope DE, 0.5 m is tied half way up. A weight 40 kg is suspended from a point F, 1.2 m from B along the ladder BA. Assuming the floor to be frictionless and neglecting the weight of the ladder, find the tension in the rope and forces exerted by the floor on the ladder. (Take $g = 9.8 m/s^2$) (Hint: Consider the equilibrium of each side of the ladder separately.)

Two particles, each of mass $m$ and speed $v$, travel in opposite directions along parallel lines separated by a distance $d$. Show that the angular momentum vector of the two particle system is the same whatever be the point about which the angular momentum is taken.

(a) Prove the theorem of perpendicular axes. (Hint : Square of the distance of a point $(x, y)$ in the $x–y$ plane from an axis through the origin and perpendicular to the plane is $x^2+y^2$).