A uniform circular disc A of radius r is made from A copper plate of thickness t and another uniform circular disc B of radius 2r is made from a copper plate of thickness t/2. 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 values of t and r.

Answer

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

### Step 1: Understand the Moment of Inertia Formula
The moment of inertia \( I \) for a uniform circular disc 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 Mass of Disc A
For disc A:
- Radius \( r_A = r \)
- Thickness \( t_A = t \)

The volume \( V_A \) of disc A is given by:
\[ V_A = \pi r_A^2 t_A = \pi r^2 t \]
The mass \( m_A \) of disc A can be calculated using the density \( \rho \):
\[ m_A = \rho V_A = \rho (\pi r^2 t) = \pi \rho r^2 t \]

### Step 3: Calculate the Moment of Inertia for Disc A
Using the mass calculated above:
\[ I_A = \frac{1}{2} m_A r_A^2 = \frac{1}{2} (\pi \rho r^2 t) r^2 = \frac{1}{2} \pi \rho t r^4 \]

### Step 4: Calculate the Mass of Disc B
For disc B:
- Radius \( r_B = 2r \)
- Thickness \( t_B = \frac{t}{2} \)

The volume \( V_B \) of disc B is given by:
\[ V_B = \pi r_B^2 t_B = \pi (2r)^2 \left(\frac{t}{2}\right) = \pi (4r^2) \left(\frac{t}{2}\right) = 2 \pi r^2 t \]
The mass \( m_B \) of disc B can be calculated using the density \( \rho \):
\[ m_B = \rho V_B = \rho (2 \pi r^2 t) = 2 \pi \rho r^2 t \]

### Step 5: Calculate the Moment of Inertia for Disc B
Using the mass calculated above:
\[ I_B = \frac{1}{2} m_B r_B^2 = \frac{1}{2} (2 \pi \rho r^2 t) (2r)^2 = \frac{1}{2} (2 \pi \rho r^2 t) (4r^2) = 4 \pi \rho t r^4 \]

### Step 6: Find the Ratio of Moments of Inertia
Now, we can find the ratio \( \frac{I_B}{I_A} \):
\[ \frac{I_B}{I_A} = \frac{4 \pi \rho t r^4}{\frac{1}{2} \pi \rho t r^4} = \frac{4}{\frac{1}{2}} = 8 \]

### Conclusion
Thus, the relationship between the moments of inertia is:
\[ I_B = 8 I_A \]