Two loops P and Q are made from a uniform wire. The radii of P and Q are `R_(1)` and `R_(2)`, respectively, and their moments of inertia about their axis of rotation are `I_(1)` and `I_(2)`, respectively. If `(I_(1))/(I_(2))=4`, then `(R_(2))/(R_(1))` is

Options

A. `4^((2)/(3))`
B. `4^((1)/(3))`
C. `4^(-(2)/(3))`
D. `4^(-(1)/(3))`

Answer

To solve the problem, we need to find the ratio of the radii \( \frac{R_2}{R_1} \) given the ratio of the moments of inertia \( \frac{I_1}{I_2} = 4 \).

### Step-by-Step Solution:

1. **Understanding the Moment of Inertia for a Loop**:
The moment of inertia \( I \) of a loop (ring) about its axis of rotation is given by the formula:
\[ I = M R^2 \]
where \( M \) is the mass of the loop and \( R \) is its radius.

2. **Finding the Mass of Each Loop**:
Since both loops are made from a uniform wire, we can express their masses in terms of their radii. The mass \( M \) of a loop can be expressed as:
\[ M = \mu \cdot L \]
where \( \mu \) is the mass per unit length and \( L \) is the length of the wire used to form the loop. The length of the wire for each loop is:
\[ L = 2 \pi R \]
Thus, the mass of loop P (with radius \( R_1 \)) is:
\[ M_1 = \mu \cdot (2 \pi R_1) = 2 \pi \mu R_1 \]
and for loop Q (with radius \( R_2 \)):
\[ M_2 = \mu \cdot (2 \pi R_2) = 2 \pi \mu R_2 \]

3. **Calculating the Moments of Inertia**:
Now substituting the masses into the moment of inertia formulas:
\[ I_1 = M_1 R_1^2 = (2 \pi \mu R_1) R_1^2 = 2 \pi \mu R_1^3 \]
\[ I_2 = M_2 R_2^2 = (2 \pi \mu R_2) R_2^2 = 2 \pi \mu R_2^3 \]

4. **Finding the Ratio of Moments of Inertia**:
Now we can find the ratio of the moments of inertia:
\[ \frac{I_1}{I_2} = \frac{2 \pi \mu R_1^3}{2 \pi \mu R_2^3} = \frac{R_1^3}{R_2^3} \]
Given that \( \frac{I_1}{I_2} = 4 \), we have:
\[ \frac{R_1^3}{R_2^3} = 4 \]

5. **Finding the Ratio of Radii**:
Taking the cube root on both sides gives:
\[ \frac{R_1}{R_2} = 4^{1/3} \]
Therefore, we can express \( \frac{R_2}{R_1} \) as:
\[ \frac{R_2}{R_1} = \frac{1}{4^{1/3}} = 4^{-1/3} \]

### Final Answer:
\[ \frac{R_2}{R_1} = 4^{-1/3} \]