A student performs exeriment with simple pendulum and measures time for `10` vibrations. If the measures the time for `100` vibrations, the error in measurement of time period will be reduced by a factor of......

Options

A. `10`
B. `90`
C. `100`
D. `1 overline(0)00`

Explanation

To solve the problem, we need to understand how the error in the measurement of the time period of a pendulum is affected by the number of vibrations measured.

### Step-by-Step Solution:

1. **Understanding the Concept**:
The error in the measurement of time period (T) is inversely proportional to the square root of the number of vibrations (N) measured. This means that as the number of vibrations increases, the relative error decreases.

2. **Initial Measurements**:
Let's denote:
- \( N_1 = 10 \) (the number of vibrations in the first measurement)
- \( T_1 \) = time measured for 10 vibrations
- \( X \) = error in the measurement of time period when measuring 10 vibrations.

3. **Final Measurements**:
Now, when the student measures the time for 100 vibrations:
- \( N_2 = 100 \) (the number of vibrations in the second measurement)
- \( T_2 \) = time measured for 100 vibrations
- \( X' \) = error in the measurement of time period when measuring 100 vibrations.

4. **Error Relationship**:
The relationship between the errors can be expressed as:
\[ \frac{X'}{X} = \frac{N_1}{N_2} \]
This indicates that the error in the second case (when measuring 100 vibrations) is reduced compared to the first case (when measuring 10 vibrations).

5. **Substituting Values**:
Substitute the values of \( N_1 \) and \( N_2 \):
\[ \frac{X'}{X} = \frac{10}{100} = \frac{1}{10} \]

6. **Calculating the Factor of Reduction**:
From the equation above, we can express \( X' \) as:
\[ X' = X \cdot \frac{1}{10} \]
This shows that the error \( X' \) in the measurement of time period when measuring 100 vibrations is \( \frac{1}{10} \) of the original error \( X \).

7. **Conclusion**:
Therefore, the error in measurement of the time period will be reduced by a factor of 10.

### Final Answer:
The error in measurement of the time period will be reduced by a factor of **10**.

---