A pendulum clock which keeps correct time at the surface of the earth is taken into a mine, then

Options

A. it keeps correct time
B. it gains time
C. it loses time
D. none ot these

Answer

To solve the problem, we need to analyze how the time period of a pendulum changes when it is taken from the surface of the Earth to a depth inside the Earth, such as in a mine.

### Step-by-Step Solution:

1. **Understanding 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.

2. **Effect of Depth on Gravity**:
When a pendulum is taken to a depth \( h \) below the Earth's surface, the acceleration due to gravity \( g \) changes. The formula for gravity at a depth \( h \) is:
\[ g' = g \left(1 - \frac{h}{R}\right) \]
where \( R \) is the radius of the Earth and \( g \) is the acceleration due to gravity at the surface.

3. **Substituting the New Gravity Value**:
When the pendulum is taken into the mine, we substitute \( g' \) into the time period formula:
\[ T' = 2\pi \sqrt{\frac{l}{g'}} \]
Substituting \( g' \):
\[ T' = 2\pi \sqrt{\frac{l}{g \left(1 - \frac{h}{R}\right)}} \]

4. **Comparing Time Periods**:
Since \( g' < g \) (because \( 1 - \frac{h}{R} < 1 \)), it follows that:
\[ T' > T \]
This means the time period of the pendulum increases when taken to a depth.

5. **Conclusion**:
As the time period increases, the pendulum will take longer to complete one oscillation, which means the clock will run slower. Therefore, the pendulum clock will lose time when taken into the mine.

### Final Answer:
The pendulum clock will run slower when taken into the mine.

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