If the length of a second's pendulum is decreased by 0.1 %, the pendulum gain or lose per day will be

Options

A. gains 43.2 s
B. loses 43.2 s
C. loses 7 s
D. loses 13.5 s

Explanation

To solve the problem of how much a second's pendulum gains or loses per day when its length is decreased by 0.1%, we can follow these steps:

### Step-by-Step Solution:

1. **Understand 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. **Differentiate the Time Period:**
Since we are interested in how a change in length affects the time period, we can differentiate \( T \) with respect to \( L \):
\[ \frac{dT}{dL} = \frac{1}{2} \cdot \frac{2\pi}{\sqrt{g}} \cdot \frac{1}{\sqrt{L}} = \frac{\pi}{\sqrt{gL}} \]

3. **Calculate the Change in Time Period:**
Given that the length \( L \) is decreased by 0.1%, we can express this change as:
\[ \Delta L = -0.001L \]
The relative change in time period \( \frac{\Delta T}{T} \) can be approximated using:
\[ \frac{\Delta T}{T} \approx \frac{1}{2} \cdot \frac{\Delta L}{L} \]
Substituting \( \Delta L = -0.001L \):
\[ \frac{\Delta T}{T} \approx \frac{1}{2} \cdot (-0.001) = -0.0005 \]

4. **Calculate the Actual Change in Time Period:**
If the original time period of a second's pendulum is \( T = 2 \) seconds (since it completes one oscillation in 2 seconds), we can find \( \Delta T \):
\[ \Delta T = \frac{\Delta T}{T} \cdot T = -0.0005 \cdot 2 = -0.001 \text{ seconds} \]

5. **Determine the Number of Oscillations in One Day:**
The number of oscillations in one day (24 hours) can be calculated as:
\[ \text{Number of oscillations} = \frac{86400 \text{ seconds}}{2 \text{ seconds/oscillation}} = 43200 \text{ oscillations} \]

6. **Calculate Total Gain/Loss in One Day:**
The total change in time over one day due to the change in time period is:
\[ \text{Total change} = \Delta T \cdot \text{Number of oscillations} = -0.001 \cdot 43200 = -43.2 \text{ seconds} \]
This indicates that the pendulum will lose 43.2 seconds in one day.

7. **Conclusion:**
Since the length of the pendulum decreased, the time period also decreased, causing the clock to gain time. Therefore, the pendulum gains 43.2 seconds per day.

### Final Answer:
The pendulum gains 43.2 seconds per day.