A pendulum clock is taken `1km` inside the earth from mean sea level. Then the pendulum clock

Options

A. loses `13.5s` per day
B. gains `13.5s` per day
C. loses `7s` per day
D. gains `7s` per day

Explanation

To solve the problem, we need to determine how the time period of a pendulum clock changes when it is taken 1 km inside the Earth. We will follow these steps:

### Step 1: Understand the formula for the time period of a pendulum
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.

### Step 2: Determine the value of \( g \) inside the Earth
As we go inside the Earth, the acceleration due to gravity \( g' \) at a distance \( x \) from the surface can be approximated by:
\[ g' = g \left(1 - \frac{x}{R}\right) \]
where \( R \) is the radius of the Earth (approximately 6378 km) and \( x \) is the distance inside the Earth (1 km in this case).

### Step 3: Calculate the new value of \( g' \)
Substituting \( x = 1 \) km and \( R = 6378 \) km into the formula:
\[ g' = g \left(1 - \frac{1}{6378}\right) = g \left(\frac{6377}{6378}\right) \]

### Step 4: Calculate the new time period \( T' \)
Using the new value of \( g' \), the new time period \( T' \) becomes:
\[ T' = 2\pi \sqrt{\frac{L}{g'}} = 2\pi \sqrt{\frac{L}{g \left(\frac{6377}{6378}\right)}} = 2\pi \sqrt{\frac{L \cdot 6378}{g \cdot 6377}} \]
This can be rewritten as:
\[ T' = \sqrt{\frac{6378}{6377}} \cdot T \]

### Step 5: Find the difference in time periods
To find the time lost, we need to calculate:
\[ T' - T = T \left(\sqrt{\frac{6378}{6377}} - 1\right) \]

### Step 6: Approximate the square root
Using a binomial approximation for small values:
\[ \sqrt{\frac{6378}{6377}} \approx 1 + \frac{1}{2} \left(\frac{1}{6377}\right) \]
Thus,
\[ T' - T \approx T \cdot \frac{1}{2 \cdot 6377} \]

### Step 7: Substitute \( T = 24 \text{ hours} = 86400 \text{ seconds} \)
Now substituting \( T \):
\[ T' - T \approx 86400 \cdot \frac{1}{2 \cdot 6377} \approx \frac{86400}{12754} \approx 6.77 \text{ seconds} \]

### Step 8: Conclusion
The pendulum clock will lose approximately 7 seconds per day when taken 1 km inside the Earth.

### Final Answer
The pendulum clock will lose 7 seconds per day.

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