If the length of a seconds pendulum is decreased by 2% find the gain or loss per day.

Explanation

To solve the problem of finding the gain or loss in time per day when the length of a seconds pendulum is decreased by 2%, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Time Period of a Pendulum**:
The time period \( P \) of a simple pendulum is given by the formula:
\[ P = 2\pi \sqrt{\frac{L}{g}} \]
where \( L \) is the length of the pendulum and \( g \) is the acceleration due to gravity.

2. **Identify the Original Length**:
For a seconds pendulum, the original time period \( P \) is 2 seconds. Therefore, we can express the original length \( L \) in terms of \( P \):
\[ 2 = 2\pi \sqrt{\frac{L}{g}} \]
Rearranging gives:
\[ L = \frac{g}{\pi^2} \]

3. **Calculate the New Length**:
If the length is decreased by 2%, the new length \( L' \) is:
\[ L' = L - 0.02L = 0.98L \]

4. **Calculate the New Time Period**:
Substitute \( L' \) into the time period formula:
\[ P' = 2\pi \sqrt{\frac{L'}{g}} = 2\pi \sqrt{\frac{0.98L}{g}} = 2\pi \sqrt{0.98} \sqrt{\frac{L}{g}} \]
Since \( P = 2\pi \sqrt{\frac{L}{g}} = 2 \), we have:
\[ P' = 2\sqrt{0.98} \]

5. **Calculate the New Time Period Value**:
The value of \( \sqrt{0.98} \) is approximately \( 0.9899 \):
\[ P' \approx 2 \times 0.9899 \approx 1.9798 \text{ seconds} \]

6. **Determine the Change in Time Period**:
The change in time period \( \Delta P \) is:
\[ \Delta P = P - P' = 2 - 1.9798 \approx 0.0202 \text{ seconds} \]

7. **Calculate the Loss in Time Over One Day**:
To find the total loss in time over one day (24 hours), we need to calculate how many cycles occur in a day:
\[ \text{Number of cycles in one day} = \frac{86400 \text{ seconds}}{2 \text{ seconds}} = 43200 \]
Therefore, the total loss in time is:
\[ \text{Total Loss} = 43200 \times 0.0202 \approx 874.56 \text{ seconds} \]

### Final Answer:
The loss in time per day is approximately **874.56 seconds**.