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 discs of same moment of inertia rotating about their regular axis passing through centre and perpendicular to the plane of disc with angular velocities ${\omega }_{1 } and {\omega }_2$. They are brought into contact face to face coinciding the axis of rotation.  The expression for loss of energy during this process is 

1. $\frac{1}{2}I{\left({\omega }_1+{\omega }_2\right)}^2$

2. $\frac{1}{4}I{\left({\omega }_1-{\omega }_2\right)}^2$

3. $I{\left({\omega }_1-{\omega }_2\right)}^2$

4. $\frac{1}{8}I{\left({\omega }_1-{\omega }_2\right)}^2$ 

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. \(L_{A} = 2 L_{B}\)
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 

1. 2:3

2. 1:5

3. 1:4

4. 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