One quarter of the disc of mass m is removed. If r be the radius of the disc, the new moment of inertia is

Options

A. `(3)/(2)mr^(2)`
B. `(mr^(2))/(2)`
C. `(3)/(8)mr^(2)`
D. None of these

Explanation

To find the new moment of inertia after removing one quarter of a disc of mass \( m \) and radius \( r \), we can follow these steps:

### Step 1: Understand the Moment of Inertia of the Whole Disc
The moment of inertia \( I \) of a full disc about an axis perpendicular to its plane and passing through its center is given by the formula:
\[ I = \frac{1}{2} m r^2 \]

### Step 2: Determine the Mass of the Removed Quarter
Since one quarter of the disc is removed, we need to calculate the mass of the removed section. The mass of the removed quarter is:
\[ \text{Mass of removed quarter} = \frac{m}{4} \]

### Step 3: Calculate the Mass of the Remaining Part
After removing one quarter of the disc, the mass of the remaining part is:
\[ \text{Remaining mass} = m - \frac{m}{4} = \frac{3m}{4} \]

### Step 4: Calculate the Moment of Inertia of the Remaining Part
The moment of inertia of the remaining part (which is now a three-quarters disc) about the same axis can be calculated. Since the radius remains the same, the moment of inertia \( I' \) of the remaining part is:
\[ I' = \frac{1}{2} \left(\frac{3m}{4}\right) r^2 \]

### Step 5: Simplify the Expression
Now, we simplify the expression for the new moment of inertia:
\[ I' = \frac{3m}{8} r^2 \]

### Final Answer
Thus, the new moment of inertia after removing one quarter of the disc is:
\[ I' = \frac{3m r^2}{8} \]

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