A uniform circular disc of radius ' R ' and mass ' M ' is rotating about an axis perpendicular to its plane and passing through its centre. A small circular part of radius R/2 is removed from the original disc as shown in the figure. Find the moment of inertia of the remaining part of the original disc about the axis as given above.

Step 1: Moment of Inertia of the Original Disc

The moment of inertia of a uniform circular disc of mass $M$ and radius $R$ about its central axis (perpendicular to its plane) is given by:

$$I_{\text{original}} = \frac{1}{2} M R^2$$

 Step 2: Moment of Inertia of the Removed Part

The removed part is a smaller disc of radius $R/2$.
Since, the original disc has uniform mass distribution, the mass of the smaller disc (proportional to its area) is:

$$M_{\text{removed}} = M \times \frac{\pi (R/2)^2}{\pi R^2} = M \times \frac{1}{4} = \frac{M}{4}$$

The moment of inertia of a smaller disc about its own center is:

$$I_{\text{removed,center}} = \frac{1}{2} M_{\text{removed}} \left(\frac{R}{2}\right)^2$$

 $$I_{\text{removed,center}} = \frac{1}{2} \times \frac{M}{4} \times \frac{R^2}{4} = \frac{1}{32} M R^2$$

However, this small disc was originally positioned at a distance $R/2$ from the center of the original disc. Using the parallel axis theorem, the moment of inertia of the removed part about the axis of the original disc is:

$$I_{\text{removed}} = I_{\text{removed,center}} + M_{\text{removed}} d^2$$

 $$I_{\text{removed}} = \frac{1}{32} M R^2 + \left(\frac{M}{4} \times \frac{R^2}{4}\right)$$

 $$I_{\text{removed}} = \frac{1}{32} M R^2 + \frac{1}{16} M R^2$$

 $$I_{\text{removed}} = \frac{1}{32} M R^2 + \frac{2}{32} M R^2 = \frac{3}{32} M R^2$$ 

Step 3: Moment of Inertia of the Remaining Part

The moment of inertia of the remaining part is:

$$I_{\text{remaining}} = I_{\text{original}} - I_{\text{removed}}$$

 $$I_{\text{remaining}} = \frac{1}{2} M R^2 - \frac{3}{32} M R^2$$

 $$I_{\text{remaining}} = \frac{16}{32} M R^2 - \frac{3}{32} M R^2$$ 

$$I_{\text{remaining}} = \frac{13}{32} M R^2$$