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

The ratio of the accelerations for a solid sphere (mass m and radius R) rolling down an incline of angle  θ without slipping and slipping down the incline without rolling will be:
1. 5:7
2. 2:3
3. 2:5
4. 7:5

A small object of uniform density rolls up a curved surface with an initial velocity v'. It reaches up to a maximum height of 3v²/4g with respect to the initial position. The object is

(1) ring

(2) solid sphere

(3) hollow sphere

(4) disc

When a mass is rotating in a plane about a fixed point, its angular momentum is directed along:

Two persons of mass 55 kg and 65 kg respectively, are at the opposite ends of a boat.The length of the boat is 3.0m and weighs 100 kg.The 55 kg man walks up to the 65 kg man and sits with him.If the boat is in still water the centre of mass of the system shifts by 

(1)3.0m                           

(2)2.3m

(3)zero                             

(4)0.75m