If two circular discs A and B are of same mass but of radii r and 2r respectively, then the moment of inertia of A is

Options

A. the same as that of B
B. twice that of B
C. four times that of B
D. one-fourth that of B

Explanation

To find the moment of inertia of two circular discs A and B with the same mass but different radii, we can follow these steps:

### Step 1: Understand the formula for moment of inertia
The moment of inertia \( I \) for a circular disc about its central axis is given by the formula:
\[ I = \frac{1}{2} m r^2 \]
where \( m \) is the mass of the disc and \( r \) is its radius.

### Step 2: Calculate the moment of inertia for disc A
For disc A, the radius is \( r \) and the mass is \( m \). Therefore, the moment of inertia \( I_A \) is:
\[ I_A = \frac{1}{2} m r^2 \]

### Step 3: Calculate the moment of inertia for disc B
For disc B, the radius is \( 2r \) and the mass is also \( m \). Therefore, the moment of inertia \( I_B \) is:
\[ I_B = \frac{1}{2} m (2r)^2 \]
Calculating this gives:
\[ I_B = \frac{1}{2} m (4r^2) = 2 m r^2 \]

### Step 4: Compare the moments of inertia
Now, we can compare \( I_A \) and \( I_B \):
\[ I_A = \frac{1}{2} m r^2 \]
\[ I_B = 2 m r^2 \]

### Step 5: Express \( I_A \) in terms of \( I_B \)
To find the relationship between \( I_A \) and \( I_B \), we can express \( I_B \) in terms of \( I_A \):
\[ I_B = 4 \left( \frac{1}{2} m r^2 \right) = 4 I_A \]
Thus, we can write:
\[ I_A = \frac{1}{4} I_B \]

### Conclusion
The moment of inertia of disc A is one-fourth that of disc B.

### Final Answer
The moment of inertia of A is one-fourth that of B.

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