A stationary horizontal disc is free to rotate about its axis. When a torque is applied on it, its kinetic energy as a function of \(\theta,\) where \(\theta\) is the angle by which it has rotated, is given as \(k\theta^2\) (where \(k\) is constant). If its moment of inertia is \(I,\) then the angular acceleration of the disc is:
1. \(\frac{k}{I} \theta\)

2. \(\frac{k}{2 I} \theta\)

3. \(\frac{k}{4 I} \theta\)

4. \(\frac{2 k}{I} \theta\)$$

A metal coin of mass \(5~\text{g}\) and radius \(1~\text{cm}\) is fixed to a thin stick \(AB\) of negligible mass as shown in the figure. The system is initially at rest. The constant torque, that will make the system rotate about \(AB\) at \(25\) rotations per second in \(5~\text{s}\), is close to:

               
1. \( 7.9 \times 10^{-6} ~\text{Nm} \)
2. \(4.0 \times 10^{-6} ~\text{Nm} \)
3. \( 2.0 \times 10^{-5} ~\text{Nm} \)
4. \( 1.6 \times 10^{-5} ~\text{Nm}\)

A solid sphere of mass \(M\) and radius \(R\) is divided into two unequal parts. The first part has a mass of \(\frac{7M}{8}\) and is converted into a uniform disc of radius \(2R\). The second part is converted into a uniform solid sphere. Let \(I_1\) be the moment of inertia of the disc about its axis and \(I_2\) be the moment of inertia of the new sphere about its axis. The ratio \(\frac{I_1}{I_2}\) is given by:
1. \(65\)
2. \(140\)
3. \(185\)
4. \(285\)

The time dependence of the position of a particle with mass \(m=2~\text{kg}\) is given by:
 \(\vec{r}(t)=(2 t \hat{i}-3 t^2 \hat{j})~\text{m}.\)
Its angular momentum, with respect to the origin, at time \(t=2~\text{s}\) is:

A thin disc of mass \(M\) and radius \(R\) has mass per unit area \(\sigma(r)=kr^2\) where \(r\) is the distance from its centre. Its moment of inertia about an axis going through its centre of mass and perpendicular to its plane is:
1. \(\frac{2MR^2}{3}\)
2. \(\frac{MR^2}{6}\)
3. \(\frac{MR^2}{3}\)
4. \(\frac{MR^2}{2}\)

A particle of mass \(m\) is moving along a trajectory given by:
\(\begin{aligned} & x=x_0+a \cos \omega_1 t \\ & y=y_0+b \sin \omega_2 t \end{aligned}\)
The torque, acting on the particle about the origin, at \(t=0\) is:
1. \( -m\left(x_0 b \omega_2^2-y_0 a \omega_1^2\right) \hat{k} \)
2. \( \text { zero } \)
3. \( +m y_0 a \omega_1^2 \hat{k} \)
4. \( m\left(-x_0 b+y_0 a\right) \omega_1^2 \hat{k}\)

Two coaxial discs, having moments of inertia \(I_1\) and \(\dfrac{I_1}{2}\) are rotating with respective angular velocities \(\omega_1\) and \(\dfrac{\omega_1}{2}\), about their common axis. They are brought in contact with each other and there after they rotate with a common angular velocity. If \(E_f\) and \(E_i\) are the final and initial total energies, then \(|E_f-E_i|\) is:

Three particles of masses \(100~\text{g}\), \(150~\text{g}\), and \(200~\text{g}\) respectively are placed at the vertices of an equilateral triangle of a side \(0.5~\text{m}\) long. What is the position of the centre of mass of three particles?

A circular disc of radius \(b\) has a hole of radius \(a\) at its centre (see figure). If the mass per unit area of the disc varies as \(\left(\frac{\sigma_0}{r}\right)\), then the radius of gyration of the disc about its axis passing through the centre is:

                             
1. \(\frac{a+b}{3}\)
2. \(\sqrt{\frac{a^2+b^2+ab}{3}}\)
3. \(\sqrt{\frac{a^2+b^2+ab}{2}}\)
4. \(\frac{a+b}{2}\)

A man (mass = \(50\) kg) and his son (mass = \(20\) kg) are standing on a frictionless surface facing each other. The man pushes his son so that he starts moving at a speed of \(0.70~\text{ms}^{-1}\) with respect to the man. The speed of the man with respect to the surface is:
1. \(0.28~\text{ms}^{-1}\)
2. \(0.47~\text{ms}^{-1}\)
3. \(0.20~\text{ms}^{-1}\)
4. \(0.14~\text{ms}^{-1}\)