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: 

1.	1.5 m	2.	2 m
3.	2.5 m	4.	3.0 m

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

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

Three particles, each of mass \(m\) gram, are situated at the vertices of an equilateral triangle ABC of side \(l\) cm (as shown in the figure). The moment of inertia of the system about a line AX, perpendicular to AB and in the plane of ABC, in gram-cm² units will be:

           
1. \(2ml^2\)
2. \(\frac{5}{4}ml^2\)
3. \(\frac{3}{2}ml^2\)
4. \(\frac{3}{4}ml^2\)

A round disc of the moment inertia ${\mathrm{I}}_2$ about its axis perpendicular to its plane and passing through its centre is placed over another disc of moment of inertia ${\mathrm{I}}_1$ rotating with an angular velocity ω about the same axis. The final angular velocity of the combination of discs is:

1. ω

2. $\frac{{\mathrm{I}}_1\omega }{{\mathrm{I}}_1+{\mathrm{I}}_2}$

3. $\frac{({\mathrm{I}}_1+{\mathrm{I}}_2)\omega }{{\mathrm{I}}_1}$

4. $\frac{{\mathrm{I}}_2\omega }{{\mathrm{I}}_1+{\mathrm{I}}_2}$