A non-uniform bar of weight \(W\) is suspended at rest by two strings of negligible weight as shown in the figure. The angles made by the strings with the vertical are \(36.9^\circ\) and  \(53.1^\circ\) respectively. The bar is \(2\) m long. The distance \(d\) of the center of gravity of the bar from its left end is:
(Take sin\(36.9^\circ=0.6\) and sin\(53.1^\circ=0.8\))

           
1. \(69\) cm
2. \(72\) cm
3. \(79\) cm
4. \(65\) cm

A car weighs \(1800~\text{kg}.\) The distance between its front and back axles is \(1.8~\text m.\) Its center of gravity is \(1.05~\text m,\) behind the front axle. The force exerted by the level ground on each front wheel and each back wheel is respectively:
1. \(2680~\text N, ~5145~\text N\)
2. \(5145~\text N, ~3675~\text N\)
3. \(5145~\text N, ~5145~\text N\)
4. \(3675~\text N, ~5145~\text N\)

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. The angular velocity of the solid sphere is:

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. The kinetic energy associated with the rotation of the cylinder is:
1. \(3000\) J
2. \(3125\) J
3. \(2528\) J
4. \(2100\) J

A child stands at the centre of a turntable with his arms outstretched. The turntable is set to rotate with an angular speed of \(40\) rev/min. How much is the angular speed of the child if he folds his hands back and thereby reduces his moment of inertia to \(2\over 5\) times the initial value?
1. \(160\) rev/min
2. \(150\) rev/min
3. \(100\) rev/min
4. \(120\) rev/min

To maintain a rotor at a uniform angular speed of \(200~\text{rad/s},\) an engine needs to transmit a torque of \(180~\text{N-m}.\) What is the power required by the engine?
1.  \(33~\text{kW}\)
2.  \(36~\text{kW}\)
3.  \(28~\text{kW}\)
4.  \(76~\text{kW}\)

A meter stick is balanced on a knife edge at its center. When two coins, each of the mass \(5\) gm are put one on top of the other at the \(12.0\) cm mark, the stick is found to be balanced at \(45.0\) cm. What is the mass of the meter stick?
1. \(66~\text{gm}\) 
2. \(56~\text{gm}\) 
3. \(76~\text{gm}\) 
4. \(79~\text{gm}\) 

A hoop of radius \(2\) m weighs \(100\) kg. It rolls along a horizontal floor so that its center of mass has a speed of \(20\) cm/s. How much work has to be done to stop it?
1. \(10\) J
2. \(9\) J
3. \(4\) J
4. \(6\) J

The oxygen molecule has a mass of \(5.30\times 10^{-26}~\text{kg}\) and a moment of inertia of \(1.94\times 10^{-46}~\text{kg m}^2\) about an axis through its center perpendicular to the lines joining the two atoms. Suppose the mean speed of such a molecule in a gas is \(500~\text{m/s}\) and that its kinetic energy of rotation is two-thirds of its kinetic energy of translation. The average angular velocity of the molecule is:
1. \(5.7\times 10^{11}~\text{rad/s}\)
2. \(5.7\times 10^{12}~\text{rad/s}\)
3. \(6.7\times 10^{11}~\text{rad/s}\)
4. \(6.7\times 10^{12}~\text{rad/s}\)

A solid cylinder rolls up an inclined plane of the angle of inclination 30°. At the bottom of the inclined plane, the centre of mass of the cylinder has a speed of 5 m/s. How far will the cylinder go up the plane?

1. 4.9 m
2. 1.3 m
3. 4.7 m
4. 3.8 m