If an experiment to determine the period of a simple pendulum of length 1m , it is attached to different spherical bobs of raddi `r_(1)` abd `r_(2)`. The two spherical bobs have uniform mass distribution . If the relative difference in the periods mass distribution.. If the relative differene in the periods is found to be `5xx10^(-4)s`, the difference in radii `|r_(1)-r_(2)|` is best given by

Options

A. 0.01cm
B. 0.1cm
C. 0.5cm
D. 1cm

Explanation

To solve the problem, we need to determine the difference in radii of two spherical bobs attached to a simple pendulum, given the relative difference in their periods.

### Step-by-Step Solution:

1. **Understand the Formula for the Period of a Simple Pendulum**:
The period \( T \) of a simple pendulum is given by the formula:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
where \( L \) is the length of the pendulum and \( g \) is the acceleration due to gravity.

2. **Identify the Length of the Pendulum**:
The length \( L \) is measured from the point of suspension to the center of mass of the bob. For two bobs with radii \( r_1 \) and \( r_2 \):
- For the first bob, the effective length is \( L + r_1 \).
- For the second bob, the effective length is \( L + r_2 \).

3. **Calculate the Periods for Both Bobs**:
- The period for the first bob \( T_1 \) is:
\[ T_1 = 2\pi \sqrt{\frac{L + r_1}{g}} \]
- The period for the second bob \( T_2 \) is:
\[ T_2 = 2\pi \sqrt{\frac{L + r_2}{g}} \]

4. **Find the Relative Difference in Periods**:
The relative difference in periods is given as:
\[ \frac{T_1 - T_2}{T_1} = \frac{5 \times 10^{-4}}{T_1} \]

5. **Use the Approximation for Small Differences**:
For small differences, we can use the approximation:
\[ T_1 - T_2 \approx \frac{dT}{dL} \cdot (r_1 - r_2) \]
where \( dT/dL \) can be derived from the period formula.

6. **Differentiate the Period Formula**:
Using the chain rule:
\[ \frac{dT}{dL} = \frac{d}{dL} \left( 2\pi \sqrt{\frac{L}{g}} \right) = \frac{\pi}{\sqrt{gL}} \]

7. **Relate the Differences in Lengths to the Radii**:
The change in period can be expressed as:
\[ T_1 - T_2 \approx \frac{\pi}{\sqrt{g}} \left( \frac{1}{\sqrt{L + r_1}} - \frac{1}{\sqrt{L + r_2}} \right) (r_1 - r_2) \]

8. **Substituting Values**:
Given that \( L = 1 \, \text{m} \) and substituting the relative difference:
\[ \frac{T_1 - T_2}{T_1} = \frac{(r_1 - r_2)}{2L} \]
leads to:
\[ \frac{5 \times 10^{-4}}{T_1} = \frac{|r_1 - r_2|}{2} \]

9. **Calculate the Difference in Radii**:
Rearranging gives:
\[ |r_1 - r_2| = 2 \times 5 \times 10^{-4} \times T_1 \]
Since \( T_1 \) is approximately \( 2\pi \sqrt{\frac{1}{g}} \) (for \( g \approx 9.81 \, \text{m/s}^2 \)):
\[ T_1 \approx 2\pi \sqrt{\frac{1}{9.81}} \approx 2 \times 3.14 \times 0.32 \approx 2.0 \, \text{s} \]
Thus,
\[ |r_1 - r_2| \approx 2 \times 5 \times 10^{-4} \times 2.0 \approx 2 \times 10^{-3} \, \text{m} = 0.002 \, \text{m} = 0.2 \, \text{cm} \]

### Final Answer:
The difference in radii \( |r_1 - r_2| \) is approximately \( 0.2 \, \text{cm} \).