The angular speed of the wheel of a vehicle is increased from \(360~\text{rpm}\) to \(1200~\text{rpm}\) in \(14\) seconds. Its angular acceleration will be:

1.	\(2\pi ~\text{rad/s}^2\)	2.	\(28\pi ~\text{rad/s}^2\)
3.	\(120\pi ~\text{rad/s}^2\)	4.	\(1 ~\text{rad/s}^2\)

Three identical spheres, each of mass \(M\), are placed at the corners of a right-angle triangle with mutually perpendicular sides equal to \(2~\text{m}\) (see figure). Taking the point of intersection of the two mutually perpendicular sides as the origin, find the position vector of the centre of mass. 
        
1. \(2(\hat{i}+\hat{j})\)
2. \(\hat{i}+\hat{j}\)
3. \(\frac{2}{3}(\hat{i}+\hat{j})\)
4. \(\frac{4}{3}(\hat{i}+\hat{j})\)

From a circular ring of mass \({M}\) and radius \(R\), an arc corresponding to a \(90^\circ\)$$sector is removed. The moment of inertia of the remaining part of the ring about an axis passing through the centre of the ring and perpendicular to the plane of the ring is \(K\) times \(MR^2\). The value of \(K\) will be:
1. \(\frac{1}{4}\)
2. \(\frac{1}{8}\)
3. \(\frac{3}{4}\)
4. \(\frac{7}{8}\)

A uniform rod of length \(200~ \text {cm}\) and mass \(500~ \text g\) is balanced on a wedge placed at \(40~ \text{cm}\) mark. A mass of \(2~\text{kg}\) is suspended from the rod at \(20~ \text{cm}\) and another unknown mass \( \text m\) is suspended from the rod at \(160~\text{cm}\) mark as shown in the figure. What would be the value of \( \mathrm{m}\) such that the rod is in equilibrium? (Take \(g=10~( \text {m/s}^2)\)

What would be the torque about the origin when a force \(3\hat{j}\) N acts on a particle whose position vector is \(2\hat{k}\) m?
1. \(6\hat{j}\) N-m
2. \(-6\hat{i}\) N-m
3. \(6\hat{k}\) N-m
4. \(6\hat{i}\) N-m

Two particles of mass \(5~\text{kg}\) and \(10~\text{kg}\) respectively are attached to the two ends of a rigid rod of length \(1~\text{m}\) with negligible mass. The centre of mass of the system from the \(5~\text{kg}\) particle is nearly at a distance of:
1. \(50~\text{cm}\)
2. \(67~\text{cm}\)
3. \(80~\text{cm}\)
4. \(33~\text{cm}\)