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 x 10^-26 kg and a moment of inertia of 1.94 x 10^-46 kg m² 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 m/s and that its kinetic energy of rotation is two-thirds of its kinetic energy of translation. Find the average angular velocity of the molecule.

1. $5.7\times {10}^{11} rad s^{-1}$

2. $5.7\times {10}^{12} rad s^{-1}$

3. $6.7\times {10}^{11} rad s^{-1}$

4.$6.7\times {10}^{12} rad s^{-1}$ 

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

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 90 cm to 20 cm. The moment of inertia of the man together with the platform may be taken to be constant and equal to 7.6 kg m². What is his new angular speed?

1. 60 rev/min
2. 57.0 rev/min
3. 58.8 rev/min
4. 60.1 rev/min

A bullet of mass \(10\) g and speed \(500\) m/s is fired into a door and gets embedded exactly at the center of the door. The door is \(1.0\) m wide and weighs \(12\) kg. It is hinged at one end and rotates about a vertical axis practically without friction. The angular speed of the door just after the bullet embeds into it is:
1. \(0.122\) rad/s
2. \(0.625\) rad/s
3. \(0.231\) rad/s
4. \(0.191\) rad/s

A disc rotating about its axis with angular speed ${\omega }_0$ is placed lightly (without any translational push) on a perfectly frictionless table. The radius of the disc is R. The linear velocities of the points A, B, and C on the disc (as shown in the figure) respectively are:
     
1. \(\omega_0 R, \omega_0 R, \frac{\omega_0 R}{2}\)
2. \(\frac{\omega_0 R}{2}, \omega_0 R, \omega_0 R\)
3. \(\omega_0 \mathrm{R}, \frac{\omega_0 \mathrm{R}}{2}, \omega_0 \mathrm{R}\)
4. \(\omega_0 R, \omega_0 R, 0\)

A cylinder of mass \(10\) kg and radius of \(15\) cm is rolling perfectly on a plane of inclination \(30^\circ.\) The coefficient of static friction \(\mu_s=0.25.\) How much is the force of friction acting on the cylinder?
1. \(16.3\) N
2. \(15.5\) N
3. \(19.1\) N
4. \(20.0\) N

Three point masses m₁, m₂ and m₃ are placed at the corners of a thin massless rectangular sheet (1.2 m x 1.0 m) as shown. Centre of mass will be located at the point

(1) (0.8, 0.6) m

(2) (0.6, 0.8) m

(3) (0.4, 0.4) m

(4) (0.5, 0.6) m

Figure shows a composite system of two uniform rods of lengths as indicated. Then the coordinates of the centre of mass of the system of rods are

(1) $\left(\frac{L}{2},\frac{2L}{3}\right)$

(2) $\left(\frac{L}{4},\frac{2L}{3}\right)$

(3) $\left(\frac{L}{6},\frac{2L}{3}\right)$

(4) $\left(\frac{L}{6},\frac{L}{3}\right)$

A man of mass m starts moving on a plank of mass M with constant velocity v with respect to plank. If the plank lles on a smooth horizontal surface, then velocity of plank with respect to ground is

(1) $\frac{Mv}{m+M}$

(2) $\frac{mv}{M}$

(3) $\frac{Mv}{m}$

(4) $\frac{mv}{m+M}$