A pendulum clock consist of aluminium rod connected to a bob. If the temperature of the surrounding becomes double , then the time period of pendulum

Options

A. Decreases
B. Remain same
C. Increases
D. May increase or decrease

Answer

To solve the problem regarding the effect of temperature on the time period of a pendulum clock made of an aluminum rod, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Time Period**:
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.

2. **Effect of Temperature on Length**:
When the temperature of the surrounding environment doubles, the aluminum rod will absorb heat. Aluminum, being a good conductor of heat, will expand due to this increase in temperature. The change in length \( \Delta L \) of the rod can be expressed using the linear expansion formula:
\[ \Delta L = L_0 \alpha \Delta T \]
where \( L_0 \) is the original length, \( \alpha \) is the coefficient of linear expansion for aluminum, and \( \Delta T \) is the change in temperature.

3. **Calculate the New Length**:
If the initial temperature is \( T_0 \) and it doubles, the new temperature \( T \) becomes \( 2T_0 \). Therefore, the change in temperature \( \Delta T = 2T_0 - T_0 = T_0 \). The new length \( L \) of the rod will be:
\[ L = L_0 + \Delta L = L_0 + L_0 \alpha T_0 = L_0(1 + \alpha T_0) \]

4. **Substitute the New Length into the Time Period Formula**:
Now, substituting the new length back into the time period formula:
\[ T' = 2\pi \sqrt{\frac{L}{g}} = 2\pi \sqrt{\frac{L_0(1 + \alpha T_0)}{g}} \]

5. **Compare the New Time Period with the Original**:
The original time period was:
\[ T = 2\pi \sqrt{\frac{L_0}{g}} \]
Since \( L \) has increased, it follows that \( T' > T \). This indicates that the time period of the pendulum increases as the length increases due to the rise in temperature.

### Conclusion:
Thus, when the temperature of the surrounding doubles, the time period of the pendulum clock will increase.