If the length of second's pendulum is decreased by 2%, how many seconds it will lose per day

Options

A. 3927s
B. 3727s
C. 3427s
D. 864s

Explanation

To solve the problem of how many seconds a second's pendulum will lose per day if its length is decreased by 2%, 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. **Identify the Original Time Period**:
For a second's pendulum, the original time period \( T \) is 2 seconds. This means that the pendulum completes one full oscillation in 2 seconds.

3. **Calculate the Change in Length**:
If the length \( L \) is decreased by 2%, we can express this change as:
\[ \Delta L = -0.02L \]
where \( L \) is the original length of the pendulum.

4. **Approximate the Change in Time Period**:
For small changes (less than 5%), we can use the approximation:
\[ \frac{\Delta T}{T} \approx \frac{1}{2} \frac{\Delta L}{L} \]
Substituting \( \Delta L = -0.02L \):
\[ \frac{\Delta T}{T} \approx \frac{1}{2} \left(-0.02\right) = -0.01 \]
This indicates that the time period will decrease by 1%.

5. **Calculate the New Time Period**:
The new time period \( T' \) can be calculated as:
\[ T' = T + \Delta T = T(1 - 0.01) = 2(1 - 0.01) = 2(0.99) = 1.98 \text{ seconds} \]

6. **Determine the Time Lost per Day**:
The time lost per oscillation is:
\[ \Delta T = T - T' = 2 - 1.98 = 0.02 \text{ seconds} \]
To find out how many oscillations occur in a day, we calculate:
\[ \text{Number of oscillations in a day} = \frac{86400 \text{ seconds}}{T} = \frac{86400}{2} = 43200 \text{ oscillations} \]

7. **Calculate Total Time Lost in a Day**:
The total time lost in a day is:
\[ \text{Total time lost} = \Delta T \times \text{Number of oscillations} = 0.02 \times 43200 = 864 \text{ seconds} \]

### Final Answer:
The second's pendulum will lose **864 seconds** per day if its length is decreased by 2%.

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