A circular disc A of radius r is made from an iron plate of thickness t and another circular disc B of radius 4r is made from an iron plate of thickness t/4. The relation between the moments of inertia `I_A and I_B` is

Options

A. `I_AgtI_B`
B. `I_A=I_B`
C. `I_AltI_B`
D. depends on the actual values of t and r

Answer

To find the relation between the moments of inertia \( I_A \) and \( I_B \) for the two circular discs, we will follow these steps:

### Step 1: Calculate the Mass of Disc A
The mass \( m_A \) of disc A can be calculated using the formula:
\[ m_A = \text{Density} \times \text{Volume} \]
The volume \( V_A \) of disc A is given by:
\[ V_A = \text{Area} \times \text{Thickness} = \pi r^2 \times t \]
Thus, the mass of disc A is:
\[ m_A = \rho \times V_A = \rho \times \pi r^2 t \]

### Step 2: Calculate the Moment of Inertia of Disc A
The moment of inertia \( I_A \) for a circular disc is given by:
\[ I_A = \frac{1}{2} m_A r^2 \]
Substituting the expression for \( m_A \):
\[ I_A = \frac{1}{2} \left( \rho \pi r^2 t \right) r^2 = \frac{1}{2} \rho \pi t r^4 \]

### Step 3: Calculate the Mass of Disc B
For disc B, the radius is \( 4r \) and the thickness is \( \frac{t}{4} \). The volume \( V_B \) of disc B is:
\[ V_B = \pi (4r)^2 \times \frac{t}{4} = \pi \times 16r^2 \times \frac{t}{4} = 4 \pi r^2 t \]
Thus, the mass of disc B is:
\[ m_B = \rho \times V_B = \rho \times 4 \pi r^2 t \]

### Step 4: Calculate the Moment of Inertia of Disc B
The moment of inertia \( I_B \) for disc B is:
\[ I_B = \frac{1}{2} m_B (4r)^2 \]
Substituting the expression for \( m_B \):
\[ I_B = \frac{1}{2} \left( \rho \times 4 \pi r^2 t \right) (16r^2) = \frac{1}{2} \times 4 \times 16 \rho \pi r^2 t \times r^2 = 32 \rho \pi t r^4 \]

### Step 5: Compare the Moments of Inertia
Now we have:
\[ I_A = \frac{1}{2} \rho \pi t r^4 \]
\[ I_B = 32 \rho \pi t r^4 \]

To compare \( I_A \) and \( I_B \):
\[ I_B = 64 I_A \]

Thus, we find that:
\[ I_A < I_B \]

### Conclusion
The relation between the moments of inertia \( I_A \) and \( I_B \) is:
\[ I_A < I_B \]