The temperature of a physical pendulum, whose time period is T, is raised by `Deltatheta.` The change in its time period is

Options

A. `1/2 alphaDelta theta`
B. `2alphe T Delta theta`
C. `1/2 alpha Delta theta`
D. 2 alpha Delata theta`

Explanation

To find the change in the time period of a physical pendulum when its temperature is raised by \( \Delta \theta \), we can follow these steps:

### Step 1: Understand the formula for the time period of a physical pendulum
The time period \( T \) of a physical pendulum is given by the formula:
\[ T = 2\pi \sqrt{\frac{l}{g}} \]
where \( l \) is the effective length of the pendulum and \( g \) is the acceleration due to gravity.

### Step 2: Consider the effect of temperature on length
When the temperature of the pendulum is raised by \( \Delta \theta \), the length \( l \) changes due to thermal expansion. The change in length \( \Delta l \) can be expressed as:
\[ \Delta l = l_0 \alpha \Delta \theta \]
where \( l_0 \) is the original length and \( \alpha \) is the coefficient of linear expansion.

### Step 3: Relate the change in length to the change in time period
The new length after the temperature increase will be:
\[ l' = l_0 + \Delta l = l_0 + l_0 \alpha \Delta \theta = l_0(1 + \alpha \Delta \theta) \]

### Step 4: Substitute the new length into the time period formula
The new time period \( T' \) can be expressed as:
\[ T' = 2\pi \sqrt{\frac{l'}{g}} = 2\pi \sqrt{\frac{l_0(1 + \alpha \Delta \theta)}{g}} \]

### Step 5: Use the binomial approximation
For small changes, we can use the binomial approximation:
\[ \sqrt{1 + x} \approx 1 + \frac{x}{2} \]
Applying this to our equation gives:
\[ T' \approx 2\pi \sqrt{\frac{l_0}{g}} \left(1 + \frac{\alpha \Delta \theta}{2}\right) = T \left(1 + \frac{\alpha \Delta \theta}{2}\right) \]

### Step 6: Calculate the change in time period
The change in time period \( \Delta T \) can be calculated as:
\[ \Delta T = T' - T = T \left(1 + \frac{\alpha \Delta \theta}{2}\right) - T = T \cdot \frac{\alpha \Delta \theta}{2} \]

### Step 7: Final expression for the change in time period
Thus, the change in the time period \( \Delta T \) is:
\[ \Delta T = \frac{\alpha \Delta \theta}{2} T \]

### Conclusion
The change in the time period of the physical pendulum when its temperature is raised by \( \Delta \theta \) is:
\[ \Delta T = \frac{\alpha \Delta \theta}{2} T \]