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) $\frac{mv}{m+M}$

A ball of mass m is thrown upward and another ball of same mass is thrown downward so as to move freely under gravity. The acceleration of centre of mass is

(1) g

(2) $\frac{g}{2}$

(3) 2g

(4) Zero

A hollow sphere of mass 1 kg and radius 10 cm is free to rotate about its diameter. If a force of 30 N is applied tangentially to it, its angular acceleration is (in rad/s²)

(1) 5000

(2) 450

(3) 50

(4) 5

Two equal and opposite forces are applied tangentially to a uniform disc of mass M and radius R as shown in the figure. If the disc is pivoted at its centre and free to rotate in its plane, the angular acceleration of the disc is

(1) $\frac{F}{MR}$

(2) $\frac{2F}{3MR}$

(3) $\frac{4F}{MR}$

(4) Zero

A wheel having moment of inertia 4 kg m² about its symmetrical axis, rotates at rate of 240 rpm about it. The torque which can stop the rotation of the wheel in one minute is

(1) $\frac{5\pi }{7}\mathrm{Nm}$

(2) $\frac{8\pi }{15}\mathrm{Nm}$

(3) $\frac{2\pi }{9}\mathrm{Nm}$

(4) $\frac{3\pi }{7}\mathrm{Nm}$