A simple pendulum made of bob of mass `m` and a metallic wire of negligible mass has time period `2 s` at `T = 0^(@)C`. If the temperature of the wire is increased and the corresponding change in its time period is plotted against its temperature, the resulting graph is a line of slope `S`. If the coefficient of linear expansion of metal is `alpha` then value of `S` is :

Options

A. `(1)/(alpha)`
B. `alpha`
C. `(alpha)/(2)`
D. `2alpha`

Answer

To find the slope \( S \) of the graph that plots the change in time period against temperature for a simple pendulum, we can follow these steps:

### Step 1: Understand the formula for the time period of a pendulum
The time 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: Consider the effect of temperature on the length of the wire
As the temperature increases, the length \( L \) of the wire will change due to thermal expansion. The change in length can be expressed as:
\[ L' = L(1 + \alpha \Delta T) \]
where \( \alpha \) is the coefficient of linear expansion and \( \Delta T \) is the change in temperature.

### Step 3: Substitute the new length into the time period formula
Substituting the new length \( L' \) into the time period formula gives:
\[ T' = 2\pi \sqrt{\frac{L(1 + \alpha \Delta T)}{g}} \]

### Step 4: Simplify the expression for the new time period
Using the approximation for small changes, we can expand the square root:
\[ T' = 2\pi \sqrt{\frac{L}{g}} \sqrt{1 + \alpha \Delta T} \approx 2\pi \sqrt{\frac{L}{g}} \left(1 + \frac{\alpha \Delta T}{2}\right) \]
Thus, we can express the new time period as:
\[ T' \approx T(1 + \frac{\alpha \Delta T}{2}) \]
where \( T \) is the original time period at \( T = 0^\circ C \).

### Step 5: Calculate the change in time period
The change in time period \( \Delta T \) can be expressed as:
\[ \Delta T = T' - T \approx T \cdot \frac{\alpha \Delta T}{2} \]
Rearranging gives:
\[ \Delta T \approx \frac{\alpha T}{2} \Delta T \]

### Step 6: Find the slope \( S \)
When plotting \( \Delta T \) against \( \Delta T \), the slope \( S \) of the line is given by:
\[ S = \frac{d(\Delta T)}{d(\Delta T)} = \frac{\alpha T}{2} \]

### Conclusion
Thus, the value of the slope \( S \) is:
\[ S = \frac{\alpha T}{2} \]
Given that \( T = 2 \, \text{s} \) at \( 0^\circ C \):
\[ S = \frac{\alpha \cdot 2}{2} = \alpha \]

### Final Answer
The value of \( S \) is \( \alpha \).

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