The temperatuire of a pendulum, the time period of which is t, is raised by `DeltaT`. The change in its time period is :

Options

A. `1/2 alpha t DeltaT`
B. `2alpha t DeltaT`
C. `1/2 alpha Delta T`
D. `2 alpha Delta T`

Answer

To solve the problem of how the time period of a pendulum changes when its temperature is raised by a certain amount, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Time Period**:
The time period \( T \) of a 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. **Consider the Effect of Temperature Change**:
When the temperature of the pendulum increases by \( \Delta T \), the length \( L \) of the pendulum will also change due to thermal expansion. The new length \( L' \) can be expressed as:
\[ L' = L(1 + \alpha \Delta T) \]
where \( \alpha \) is the coefficient of linear thermal expansion.

3. **Substitute the New Length into the Time Period Formula**:
The new time period \( T' \) after the temperature change can be written as:
\[ T' = 2\pi \sqrt{\frac{L'}{g}} = 2\pi \sqrt{\frac{L(1 + \alpha \Delta T)}{g}} \]

4. **Factor Out the Original Time Period**:
We can factor out the original time period \( T \):
\[ T' = 2\pi \sqrt{\frac{L}{g}} \sqrt{1 + \alpha \Delta T} = T \sqrt{1 + \alpha \Delta T} \]

5. **Use the Binomial Approximation**:
For small values of \( \alpha \Delta T \), we can use the binomial approximation:
\[ \sqrt{1 + x} \approx 1 + \frac{x}{2} \]
Thus, we can approximate:
\[ T' \approx T \left(1 + \frac{\alpha \Delta T}{2}\right) \]

6. **Calculate the Change in Time Period**:
The change in time period \( \Delta T \) can be calculated as:
\[ \Delta T = T' - T \approx T \left(1 + \frac{\alpha \Delta T}{2}\right) - T = T \cdot \frac{\alpha \Delta T}{2} \]

7. **Final Result**:
Therefore, the change in the time period of the pendulum when the temperature is raised by \( \Delta T \) is:
\[ \Delta T = \frac{1}{2} \alpha T \Delta T \]

### Summary:
The change in the time period of the pendulum due to a temperature increase of \( \Delta T \) is given by:
\[ \Delta T = \frac{1}{2} \alpha T \Delta T \]