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) $\sqrt{m_1{m}_2{l}^2}$ 

From a disc of radius R and mass M, a circular hole of diameter R, whose rim passes through the centre is cut. What is the moment of inertia of the remaining part of the disc about a perpendicular axis, passing through the centre ?

(1) $13 M{R}^2/32$       

(2) $11 M{R}^2/32$

(3) $9 M{R}^2/32$         

(4) $15 M{R}^2/32$

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

A force \(\vec{F}=\alpha\hat i+3\hat j+6\hat k\) is acting at a point \(\vec{r}=2\hat i-6\hat j-12\hat k.\) The value of \(\alpha\) for which angular momentum is conserved about the origin is:
1. \(-1\)
2. \(2\)
3. zero
4. \(1\)

An automobile moves on a road with a speed of 54 km h^-1. The radius of its wheels is 0.45 m and the moment of inertia of the wheel about its axis of rotation is 3 kg m². If the vehicle is brought to rest in 15 s, the magnitude of average torque transmitted by its brakes to the wheel is 

(1)6.66 kgm²s^-2

(2)8.58 kgm²s^-2

(3)10.86 kgm²s^-2

(4)2.86 kgm²s^-2

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

(1)25N

(2)50N

(3)78.5N

(4)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 is

(1) 5:7

(2) 2:3

(3) 2:5

(4) 7:5

A rod PQ of mass M and length L is hinged at end P. The rod is kept horizontal by a muscle string tied to point Q as shown in the figure. When the string is cut, the initial angular acceleration of the rod is -

(1)3g/2L

(2)g/L

(3)2g/L

(4)2g/3L

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

1. a line perpendicular to the plane of rotation

2. the line making an angle of $45°$ to the plane of rotation

3. the radius

4. the tangent to the orbit