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

Options

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

Explanation

To solve the problem of finding the difference in radii of the spherical bobs based on the given relative difference in their periods, we can follow these steps:

### Step 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.

### Step 2: Differentiate the period with respect to the length
Taking the logarithm of both sides, we have:
\[ \log T = \log(2\pi) + \frac{1}{2} \log L - \frac{1}{2} \log g \]
Differentiating both sides gives:
\[ \frac{dT}{T} = \frac{1}{2} \frac{dL}{L} \]

### Step 3: Relate the change in period to the change in length
From the differentiation, we can express the relative change in the period \( \frac{\Delta T}{T} \) in terms of the relative change in length \( \frac{\Delta L}{L} \):
\[ \frac{\Delta T}{T} = \frac{1}{2} \frac{\Delta L}{L} \]

### Step 4: Substitute the known values
We know that the relative difference in the periods is given as:
\[ \frac{\Delta T}{T} = 5 \times 10^{-4} \]
Since the length \( L \) of the pendulum is 1 m, we can substitute \( L = 1 \):
\[ 5 \times 10^{-4} = \frac{1}{2} \frac{\Delta L}{1} \]

### Step 5: Solve for \( \Delta L \)
Rearranging the equation gives:
\[ \Delta L = 2 \times 5 \times 10^{-4} = 10 \times 10^{-4} = 10^{-3} \text{ m} \]

### Step 6: Convert to centimeters
Since \( 1 \text{ m} = 100 \text{ cm} \):
\[ \Delta L = 10^{-3} \text{ m} = 0.1 \text{ cm} \]

### Conclusion
Thus, the difference in radii \( |r_1 - r_2| \) is \( 0.1 \text{ cm} \).

### Final Answer
The difference in radii \( |r_1 - r_2| \) is \( 0.1 \text{ cm} \).

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