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 of how the time period of a pendulum clock changes when the temperature of the surrounding environment doubles, we can follow these steps:

### Step 1: Understand the Time Period Formula
The time period \( T \) of a simple pendulum is given by the formula:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
where:
- \( T \) is the time period,
- \( L \) is the length of the pendulum,
- \( g \) is the acceleration due to gravity.

### Step 2: Consider the Effect of Temperature on Length
When the temperature of the surrounding environment increases, the aluminum rod of the pendulum will undergo linear expansion. The length \( L \) of the rod will increase due to this expansion.

The linear expansion of a material can be expressed as:
\[ \Delta L = L_0 \alpha \Delta T \]
where:
- \( \Delta L \) is the change in length,
- \( L_0 \) is the original length,
- \( \alpha \) is the coefficient of linear expansion for aluminum,
- \( \Delta T \) is the change in temperature.

If the temperature doubles, the change in temperature \( \Delta T \) is significant enough to cause a noticeable increase in length \( L \).

### Step 3: Analyze the New Length
Let’s denote the new length after the temperature increase as \( L' \). Since the length increases, we can say:
\[ L' > L \]

### Step 4: Substitute the New Length into the Time Period Formula
The new time period \( T' \) can be expressed as:
\[ T' = 2\pi \sqrt{\frac{L'}{g}} \]
Since \( L' > L \), it follows that:
\[ \sqrt{L'} > \sqrt{L} \]
Thus, we can conclude that:
\[ T' > T \]

### Step 5: Conclusion
The time period of the pendulum increases when the temperature of the surrounding environment doubles.

### Final Answer
The time period of the pendulum increases.

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