A body of mass (m) is projected with velocity v₀ at an angle of projection $\mathrm{\theta }$. The magnitude of angular momentum about origin after time t from projection is :

1. Increase with time

2. decrease with time

3. independent of time

4. none of these

A body of mass m slides down an incline and reaches the bottom with a velocity v. If the same mass were in the form of a ring which rolls down this incline, the velocity of the ring at bottom would have been

(1) v

(2) $\sqrt{2} v$

(3) $\left(1/\sqrt{2}\right)v$

(4) $\left(\sqrt{2/5}\right)v$

A rope is wound around a hollow cylinder of mass 3 kg and radius 40cm. What is the angular acceleration of the cylinder,if the rope is pulled with a force of 30 N?

(1) 25$m/s^2$

(2) 0.25 $rad/s^2$

(3) 25 $rad/s^2$

(4) 5 $m/s^2$

Two rotating bodies A and B of masses m and 2m with moments of inertia $I_A$ and $I_B(I_B>I_A)$ have equal kinetic energy of rotation. If $L_{A }$and $L_B$ be their angular momenta respectively, then 

(1) $L_A=\frac{L_B}{2}$

(2) $L_A=2L_B$

(3) $L_B>L_A$

(4) $L_A>L_B$

A solid sphere of mass m and radius R is rotating about its diameter. A soild cyclinder of the same mass and same radius is also rotating about its geometrical axis with an angular speed twice that of the sphere. The ratio of their kinetic energies of rotation $\left(E_{sphere}/E_{cylinder }\right)$ will be 

(a) 2:3

(b) 1:5

(c) 1:4

(d) 3:1

A light rod of length l has two masses $m_1 and m_2$ attached to its two ends. The moment of inertia of the system about an axis perpendicular to the rod and passing through the centre of mass is 

(1) $\frac{m_1{m}_2}{m_1+m_2}{l}^2$

(2)$\frac{m_1+m_2}{\sqrt{m_1{m}_2{l}^2}}{l}^2$

(3)$\left(m_1+m_2\right)l^2$

(4) $\sqrt{m_1{m}_2{l}^2}$ 

A rod of weight w is supported by two parallel knife edges A and B and is in equilibrium in a horizontal position. The knives are at a distance d from each other. The centre of mass of the rod is at distance x from A. The normal reaction on A is

(1)$\frac{\mathrm{wx}}{\mathrm{d}}$

(2)$\frac{\mathrm{wd}}{\mathrm{x}}$

(3)w(d-x)/x

(4)$\frac{w(d-x)}{d}$

Three identical spherical shells, each of mass m and radius r, are placed as shown in the figure. Consider an axis XX’, which is touching two shells and passes through the diameter of the third shell. The moment of inertia of the system consisting of these three spherical shells about the XX' axis is:
    
1.$\frac{11}{5}$mr²

2. 3mr²

3. $\frac{16}{5}$mr²

4. 4mr²

A force F=$a\hat{i}$$+3\hat{j}$$+6\hat{k}$ is acting at a point r=$2\hat{i}$-$6\hat{j}$-12$\hat{k}$. The value of α for which angular momentum about origin is conserved is

(1) -1

(2) 2

(3) zero

(4) 1

A solid cylinder of mass 50 kg and radius 0.5 m is free to rotate about the horizontal axis.A massless string is wound round the cylinder with one end attached to it and other hanging freely.Tension in the string required to produce an angular acceleration of 2 rev/ s² is

(a)25N

(b)50N

(c)78.5N

(d)157N