A particle of mass \(5m\) at rest suddenly breaks on its own into three fragments. Two fragments of mass \(m\) each move along mutually perpendicular directions with speed \(v\) each. The energy released during the process is:

1.	\(\dfrac{3}{5}mv^2\)	2.	\(\dfrac{5}{3}mv^2\)
3.	\(\dfrac{3}{2}mv^2\)	4.	\(\dfrac{4}{3}mv^2\)

A solid cylinder of mass 2 kg and radius 50 cm rolls up an inclined plane of angle of inclination ${30}^{°}$. The centre of mass of the cylinder has a speed of 4 m/s. The distance travelled by the cylinder on the inclined surface will be $[take$ $g=$ $10$ $m/s^2]\; :$

1. 2.2 m

2. 1.6 m

3. 1.2 m

4. 2.4 m

A particle starting from rest moves in a circle of radius \(r\). It attains a velocity of \(v_0~\text{m/s}\) on completion of \(n\) rounds. Its angular acceleration will be:
1. \( \dfrac{v_0}{n} ~\text{rad} / \text{s}^2\)
2. \( \dfrac{v_0^2}{2 \pi {nr}^2}~ \text{rad} / \text{s}^2 \)
3. \( \dfrac{v_0^2}{4 \pi {n}{r}^2}~ \text{rad} / \text{s}^2 \)
4. \( \dfrac{v_0^2}{4 \pi {nr}} ~\text{rad} / \text{s}^2 \)

An object flying in the air with velocity \((20 \hat{i}+25 \hat{j}-12 \hat{k})\) suddenly breaks into two pieces whose masses are in the ratio of \(1:5.\) The smaller mass flies off with a velocity \((100 \hat{i}+35 \hat{j}+8 \hat{k})\)$\mathrm{}$. The velocity of the larger piece will be:
1. \( 4 \hat{i}+23 \hat{j}-16 \hat{k}\)
2. \( -100 \hat{i}-35 \hat{j}-8 \hat{k} \)
3. \( 20 \hat{i}+15 \hat{j}-80 \hat{k} \)
4. \( -20 \hat{i}-15 \hat{j}-80 \hat{k}\)

A disc of radius \(2~\text{m}\) and mass \(100~\text{kg}\) rolls on a horizontal floor. Its centre of mass has a speed of \(20~\text{cm/s}\). How much work is needed to stop it?
1. \(1~\text{J}\)
2. \(3~\text{J}\)
3. \(30~\text{J}\)
4. \(2~\text{J}\)

A solid cylinder of mass \(2~\text{kg}\) and radius \(4~\text{cm}\) is rotating about its axis at the rate of \(3~\text{rpm}.\) The torque required to stop after \(2\pi\) revolutions is:
1. \(2\times 10^6~\text{N-m}\)
2. \(2\times 10^{-6}~\text{N-m}\) 
3. \(2\times 10^{-3}~\text{N-m}\) 
4. \(12\times 10^{-4}~\text{N-m}\)

The moment of inertia of a uniform circular disc of radius \(R\) and mass \(M\) about an axis touching the disc at its diameter and normal to the disc is:
1. $M{R}^2$

2. $\frac{2}{5}M{R}^2$

3. $\frac{3}{2}M{R}^2$

4. $\frac{1}{2}M{R}^2$

A particle of mass \(m\) moves in the XY plane with a velocity \(v\) along the straight line AB. If the angular momentum of the particle with respect to the origin \(O\) is \(L_A\) when it is at \(A\) and \(L_B\) when it is at \(B,\) then: 
         

A wheel has an angular acceleration of \(3.0\) rad/s² and an initial angular speed of \(2.0\) rad/s. In a time of \(2\) s, it has rotated through an angle (in radians) of:

The ratio of the radii of gyration of a circular disc to that of a circular ring, each of the same mass and radius, around their respective axes is: