List of top Physics Questions on Moment Of Inertia asked in IIT JAM PH

A thin circular disc lying in the \( xy \)-plane has a surface mass density \[ \sigma(r) = \begin{cases} \sigma_0\left(1 - \frac{r^2}{R^2}\right), & \text{if } r \le R \\ 0, & \text{if } r > R \end{cases} \] where \( r \) is the distance from its center. Its moment of inertia about the z-axis passing through its center is:
A particle, initially at the origin in an inertial frame \( S \), has a constant velocity \( V\hat{i} \). Frame \( S' \) is rotating about the \( z \)-axis with angular velocity \( \omega \) (anticlockwise). The coordinate axes of \( S' \) coincide with those of \( S \) at \( t = 0 \). The velocity of the particle \((V'_x, V'_y)\) in the \( S' \) frame, at \( t = \frac{\pi}{2\omega} \), is:
The moment of inertia of a solid sphere (radius \( R \) and mass \( M \)) about the axis which is at a distance of \( \frac{R}{2} \) from the center is:

Consider a uniform thin circular disk of radius \( R \) and mass \( M \). A concentric square of side \( R/2 \) is cut out from the disk (see figure). What is the moment of inertia of the resultant disk about an axis passing through the centre of the disk and perpendicular to it? 

Consider a uniform thin circular disk of radius \( R \) and mass \( M \). A concentric square of side \( R/2 \) is cut out from the disk (see figure). What is the moment of inertia of the resultant disk about an axis passing through the centre of the disk and perpendicular to it?