A disc of radius r is removed from the centre of a disc of mass M and radius R. The M.I. About the axis through the centre and normal to the plane will be:

Options

A. `M(R^(2)-r^(2))//2`
B. `M(R^(2)+r^(2))//2`
C. `M(R^(4)-r^(4))//2R^(2)`
D. `M(R^(2))//2`

Answer

To find the moment of inertia of a disc with a smaller disc removed from its center, we can follow these steps:

### Step 1: Moment of Inertia of the Whole Disc
The moment of inertia \( I \) of a solid disc of mass \( M \) and radius \( R \) about an axis through its center and normal to its plane is given by the formula:
\[ I = \frac{1}{2} M R^2 \]

### Step 2: Moment of Inertia of the Smaller Disc
Next, we need to find the moment of inertia of the smaller disc that is being removed. Let the radius of the smaller disc be \( r \) and its mass be \( m' \). The mass \( m' \) can be calculated using the area density of the larger disc. The area of the larger disc is \( \pi R^2 \) and the area of the smaller disc is \( \pi r^2 \). The mass per unit area \( \sigma \) of the larger disc is:
\[ \sigma = \frac{M}{\pi R^2} \]
Thus, the mass \( m' \) of the smaller disc is:
\[ m' = \sigma \cdot \text{Area of smaller disc} = \frac{M}{\pi R^2} \cdot \pi r^2 = \frac{M r^2}{R^2} \]

The moment of inertia \( I' \) of the smaller disc about the same axis is given by:
\[ I' = \frac{1}{2} m' r^2 = \frac{1}{2} \left(\frac{M r^2}{R^2}\right) r^2 = \frac{1}{2} \frac{M r^4}{R^2} \]

### Step 3: Net Moment of Inertia After Removing the Smaller Disc
The net moment of inertia \( I_{\text{net}} \) after removing the smaller disc from the larger disc is:
\[ I_{\text{net}} = I - I' = \frac{1}{2} M R^2 - \frac{1}{2} \frac{M r^4}{R^2} \]
Factoring out \( \frac{1}{2} M \):
\[ I_{\text{net}} = \frac{1}{2} M \left( R^2 - \frac{r^4}{R^2} \right) \]
To combine the terms, we can write:
\[ I_{\text{net}} = \frac{1}{2} M \left( \frac{R^4 - r^4}{R^2} \right) \]

### Final Result
Thus, the moment of inertia about the axis through the center and normal to the plane after removing the smaller disc is:
\[ I_{\text{net}} = \frac{M}{2} \cdot \frac{R^4 - r^4}{R^2} \]