In the \(\mathrm{HCl}\) molecule, the separation between the nuclei of the two atoms is about \(1.27~\mathring{\text A}~(1~\mathring{\text A}=10^{10}~\text m).\) Then the approximate location of the CM of the molecule is:
(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).

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, then the speed of the centre of mass of the (trolley + child) system:
1. decreases
2. increases
3. remains unchanged
4. none of these

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 kg. The distance between its front and back axles is 1.8 m. Its center of gravity is 1.05 m behind the front axle. The force exerted by the level ground on each front wheel and each back wheel is respectively:
1. 2680 N, 5145 N
2. 5145 N, 3675 N
3. 5145 N, 5145 N
4. 3675 N, 5145 N

Given the moment of inertia of a disc of mass \(M\) and radius \(R\) about any of its diameters to be \(\dfrac{MR^{2}}{4},\) then the moment of inertia about an axis normal to the disc passing through a point on its edge is:
1. \(\dfrac{3}{2}MR^{2}\)
2. \(\dfrac{1}{4}MR^{2}\)
3. \(\dfrac{2}{5}MR^{2}\)
4. \(\dfrac{7}{5}MR^{2}\)

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 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

A rope of negligible mass is wound around 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?
(Assume that there is no slipping.)
1. \(21\) rad/s²
2. \(24\) rad/s²
3. \(20\) rad/s²
4. \(25\) rad/s²

To maintain a rotor at a uniform angular speed of \(200\) rad s^–1, an engine needs to transmit a torque of \(180\) N-m. What is the power required by the engine?
1.  \(33\) kW
2.  \(36\) kW
3.  \(28\) kW
4.  \(76\) 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. \(79~\text{gm}\)