A square shaped hole of side \(l=\dfrac{a}{2}\) is carved out at a distance \(d=\dfrac{a}{2}\) from the centre '\(O\)' of a uniform circular disk of radius \(a\). If the distance of the centre of mass of the remaining portion from \(O\) is \(-\dfrac{a}{x}\) value of \(x\) (to the nearest integer) is:

       
1. \(12\)
2. \(23\)
3. \(45\)
4. \(76\)

Moment of inertia of a cylinder of mass \(M\), length \(L\) and radius \(R\) about an axis passing through its centre and perpendicularto the axis of the cylinder is \(I=M\left(\frac{R^2}{4}+\frac{L^2}{12}\right) \) .If such a cylinder is to be made for a given mass of a material, the ratio \(L/R\) for it to have minimum possible \(I\) is:
1. \( \frac{2}{3} \)
2. \( \frac{3}{2} \)
3. \(\sqrt{\frac{2}{3}} \)
4. \( \sqrt{\frac{3}{2}}\)

A person with a mass of \(80~\text{kg}\) is standing on the rim of a circular platform with a mass of \(200~\text{kg}\) and rotating about its axis at a speed of \(5\) revolutions per minute (rpm). As the person moves toward the centre of the platform, what will be the platform's new rotational speed (in rpm) once the person reaches its centre?
1. \(3\)
2. \(6\)
3. \(9\)
4. \(12\)

An massless equilateral triangle \(EFG\) of side \(a\) (As shown in figure) has three particles of mass \(m\) situated at its vertices. The moment of inertia of the system about the line \(EX\) perpendicular to \(EG\) in the plane of \(EFG\) is \(\frac{N}{20} m a^2\) where \(N\) is an integer. The value of \(N\) is:

 
1. \(10\)
2. \(15\)
3. \(20\)
4. \(25\)

A wheel of moment of inertia \(I\) rotates freely with angular speed \(\omega\) on a shaft, whose inertia is negligible. Another wheel of moment of inertia \(3I,\) initially at rest, is suddenly coupled to the same shaft. What is the fractional loss of kinetic energy of the system after the wheels are coupled?

A force \(\vec{F}=\hat{i}+2\hat{j}+3\hat{k}~\text{N}\) acts at a point \(\hat{4i}+3\hat{j}-\hat{k}~\text{m}\). Let the magnitude of the torque about the point \(\hat{i}+2\hat{j}+\hat{k}~\text{m}\) be \(\sqrt{x}~\text{N-m}\). The value of \(x\) is:
1. \(145\)
2. \(195\)
3. \(245\)
4. \(295\)

A thin rod of mass \(0.9 ~\text{kg}\) and length \(1~\text{m}\) is suspended, at rest, from one end so that it can freely oscillate in the vertical plane. A particle of mass \(0.1~\text{kg}\) moving in a straight line with velocity \(80~\text{m/s}\) hits the rod at its bottom most point and sticks to it (see figure). The angular speed (in rad/s) of the rod immediately after the collision will be:

  
1. \(20\)
2. \(40\)
3. \(60\)
4. \(80\)

\(ABC\) is a plane lamina of the shape of an equilateral triangle. \(D\), \(E\) are mid points of \(AB\), \(AC\) and \(G\) is the centroid of the lamina. The moment of inertia of the lamina about an axis passing through \(G\) and perpendicular to the plane \(ABC\) is \(I_0\). If part \(ADE\) is removed, the moment of inertia of the remaining part about the same axis is \(\frac{N I_0}{16}\) where \(N\) is an integer. Value of \(N\) is:

1. \(3\)
2. \(20\)
3. \(11\)
4. \(38\)

A circular disc of mass \(M\) and radius \(R\) is rotating about its axis with angular speed \(\omega_1\). If another stationary disc having radius \(\frac{R}{2}\) and same mass \(M\) is dropped co-axially on to the rotating disc. Gradually both discs attain constant angular speed \(\omega_2\). The energy lost in the process is \(p\%\) of the initial energy. Value of \(p\) is:
1. \(10\)
2. \(20\)
3. \(30\)
4. \(40\)

A sphere of radius '\(a\)' and mass '\(m\)' rolls along a horizontal plane with constant speed \(v_0\). It encounters an inclined plane at angle \(\theta \) and climbs upward. Assuming that it rolls without slipping, how far up the sphere will travel ?

   
1. \( \frac{7 v_0^2}{10 g \sin \theta} \)
2. \(\frac{v_0^2}{5 g \sin \theta} \)
3. \(\frac{2}{5} \frac{v_0^2}{g \sin \theta} \)
4. \(\frac{v_0^2}{2 \operatorname{gsin} \theta} \)