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. \(\frac{v_0^2}{2 \operatorname{gsin} \theta} \)

Seven identical circular planar disks, each of mass \(M\) and radius \(R\) are welded symmetrically as shown. The moment of inertia of the arrangement about the axis normal to the plane and passing through the point \(P\) is:

          
1. \(\frac{19}{2}{MR}^2\)
2. \(\frac{55}{2}{MR}^2\)
3. \(\frac{73}{2}{MR}^2\)
4. \(\frac{181}{2}{MR}^2\)

From a uniform circular disc of radius \(R\) and mass \(9M\), a small disc of radius \(\frac{R}{3}\) is removed as shown in the figure. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through centre of the disc is:

   
1. \( 4{MR}^2 \)
2. \( \frac{40}{9}{MR}^2 \)
3. \( 10{MR}^2 \)
4. \( \frac{37}{9}{MR}^2\)

The moments of inertia of four bodies, all having the same mass and radius, are reported as follows:

Which of the following relationships is correct?
1. \(I_1+I_3<I_2+I_4\)
2. \(I_1+I_2= I_3+\frac{5}{2}I_4\)
3. \(I_1=I_2=I_3>I_4\)
4. \(I_1=I_2=I_3<I_4\)