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 `mu`

Options

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

Answer

To solve the problem, we need to find the value of \( \mu \) given that the moment of inertia of two discs with radii \( R \) and \( \mu R \) are in the ratio \( 1:16 \).

### Step-by-Step Solution:

1. **Understanding the Moment of Inertia of a Disc**:
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. **Mass of the Discs**:
Since both discs are made of the same material and have the same thickness, we can express their masses in terms of their areas and the density of the material. The mass \( m \) of a disc can be expressed as:
\[ m = \sigma \cdot A \]
where \( \sigma \) is the mass per unit area and \( A \) is the area of the disc. The area \( A \) of a disc with radius \( r \) is:
\[ A = \pi r^2 \]
Therefore, the mass of the first disc (radius \( R \)) is:
\[ m_1 = \sigma \cdot \pi R^2 \]
and the mass of the second disc (radius \( \mu R \)) is:
\[ m_2 = \sigma \cdot \pi (\mu R)^2 = \sigma \cdot \pi \mu^2 R^2 \]

3. **Calculating the Moments of Inertia**:
Now we can calculate the moments of inertia for both discs:
- For the first disc:
\[ I_1 = \frac{1}{2} m_1 R^2 = \frac{1}{2} (\sigma \cdot \pi R^2) R^2 = \frac{1}{2} \sigma \pi R^4 \]
- For the second disc:
\[ I_2 = \frac{1}{2} m_2 (\mu R)^2 = \frac{1}{2} (\sigma \cdot \pi \mu^2 R^2) (\mu R)^2 = \frac{1}{2} \sigma \pi \mu^2 R^4 \]

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 \( I_2 \):
\[ \frac{\frac{1}{2} \sigma \pi R^4}{\frac{1}{2} \sigma \pi \mu^2 R^4} = \frac{1}{16} \]
The constants \( \frac{1}{2} \sigma \pi R^4 \) cancel out, leading to:
\[ \frac{1}{\mu^2} = \frac{1}{16} \]

5. **Solving for \( \mu \)**:
Rearranging the equation gives:
\[ \mu^2 = 16 \]
Taking the square root of both sides:
\[ \mu = 4 \]

### Final Answer:
The value of \( \mu \) is \( 4 \).