Two disc made of same material and same thickness having radius `R` and `mu R`.Their moment of inertia about their own axis are in ratio `1: 16`. Calculate the value of `alpha`

Options

A. `2`
B. `1/2`
C. `1`
D. `1/4`

Answer

To solve the problem, we need to find the value of \( \alpha \) given the ratio of the moments of inertia of two discs made of the same material and thickness, with radii \( R \) and \( \mu R \) respectively.

### Step-by-Step Solution:

1. **Understanding Moment of Inertia**:
The moment of inertia \( I \) of a disc about its own 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.

2. **Calculating Mass of the Discs**:
The mass \( m \) of a disc can be calculated using the formula:
\[ m = \text{density} \times \text{volume} \]
The volume \( V \) of a disc is given by:
\[ V = \text{Area} \times \text{Thickness} = \pi r^2 T \]
Therefore, the mass of the first disc (radius \( R \)) is:
\[ m_1 = \rho \cdot \pi R^2 T \]
And for the second disc (radius \( \mu R \)):
\[ m_2 = \rho \cdot \pi (\mu R)^2 T = \rho \cdot \pi \mu^2 R^2 T \]

3. **Calculating Moments of Inertia**:
For the first disc:
\[ I_1 = \frac{1}{2} m_1 R^2 = \frac{1}{2} \left(\rho \cdot \pi R^2 T\right) R^2 = \frac{1}{2} \rho \pi T R^4 \]
For the second disc:
\[ I_2 = \frac{1}{2} m_2 (\mu R)^2 = \frac{1}{2} \left(\rho \cdot \pi \mu^2 R^2 T\right) (\mu R)^2 = \frac{1}{2} \rho \pi T \mu^4 R^4 \]

4. **Setting Up the Ratio**:
According to the problem, the ratio of the moments of inertia is given as:
\[ \frac{I_1}{I_2} = \frac{1}{16} \]
Substituting the expressions for \( I_1 \) and \( I_2 \):
\[ \frac{\frac{1}{2} \rho \pi T R^4}{\frac{1}{2} \rho \pi T \mu^4 R^4} = \frac{1}{16} \]

5. **Simplifying the Ratio**:
The constants \( \frac{1}{2} \rho \pi T R^4 \) cancel out:
\[ \frac{1}{\mu^4} = \frac{1}{16} \]

6. **Solving for \( \mu \)**:
Taking the reciprocal gives:
\[ \mu^4 = 16 \]
Taking the fourth root:
\[ \mu = 2 \]

### Final Answer:
Thus, the value of \( \alpha \) is:
\[ \alpha = 2 \]