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 \(g=10~( \text {m/s}^2)\)

A thin circular ring \(\mathrm{M}\) and radius \(\mathrm{r}\) is rotating about its axis with a constant angular velocity ω. Four objects, each of mass m, are kept gently to the opposite ends of two perpendicular diameters of the ring. The angular velocity of the ring will be:

1. $\frac{\mathrm{M\omega }}{4\mathrm{m}}$

2. $\frac{\mathrm{M\omega }}{\mathrm{M}+4\mathrm{m}}$

3. $\frac{(\mathrm{M}+4\mathrm{m})\mathrm{\omega }}{\mathrm{M}}$

4. $\frac{(\mathrm{M}+4\mathrm{m})\mathrm{\omega }}{\mathrm{M}+4\mathrm{m}}$

If a rod of length 3 m with its mass acting per unit length, is directly proportional to distance x from one of its ends, then its centre of gravity from that end will be at: 

When a stick is released (as shown in the figure below), its free end velocity when it strikes the ground is:

1. 4.2 m/s
2. 1.4 m/s
3. 2.8 m/s
4. $\sqrt{6}$ m/s

For a body, with angular velocity \( \vec{\omega }=\hat{i}-2\hat{j}+3\hat{k}\)  and radius vector \( \vec{r }=\hat{i}+\hat{j}++\hat{k},\)  its velocity will be:
1. \(-5\hat{i}+2\hat{j}+3\hat{k}\)
2. \(-5\hat{i}+2\hat{j}-3\hat{k}\)
3. \(-5\hat{i}-2\hat{j}+3\hat{k}\)
4. \(-5\hat{i}-2\hat{j}-3\hat{k}\)

A disc is rotating with angular speed \(\omega.\) If a child sits on it, what is conserved here?
1. Linear momentum
2. Angular momentum
3. Kinetic energy
4. Potential energy

A circular disc is to be made by using iron and aluminium so that it acquires a maximum moment of inertia about its geometrical axis. It is possible with: 

Consider a system of two particles having masses m₁ and m₂. If the particle of mass m₁ is pushed towards the mass centre of particles through a distance 'd', by what distance would the particle of mass m₂ move so as to keep the mass centre of particles at the original position ?

1. $\frac{{\mathrm{m}}_1}{{\mathrm{m}}_2}\mathrm{d}$

2. d

3. $\frac{{\mathrm{m}}_2}{{\mathrm{m}}_1}$

4. $\frac{{\mathrm{m}}_1}{{\mathrm{m}}_1+{\mathrm{m}}_2}\mathrm{d}$

A wheel having a moment of inertia of \(2\) kg–m² about its vertical axis rotates at the rate of \(60\) rpm about the axis. The torque which can stop the wheel's rotation in one minute would be:
1. \(\frac{\pi }{12}\) N-m

2.  \(\frac{\pi }{15}\) N-m

3.  \(\frac{\pi }{18}\) N-m

4.  \(\frac{2\pi }{15}\) N-m