What will be the time of oscialltion, If the length of a second pendulum is onoe third ?

Options

A. `(2//3)s`
B. `(1//sqrt(3))s`
C. `(2//sqrt(3))s`
D. `4//sqrt(3)s`

Explanation

To find the time of oscillation of a pendulum whose length is one third of that of a second pendulum, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for 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:
- \( T \) is the time period,
- \( L \) is the length of the pendulum,
- \( g \) is the acceleration due to gravity.

2. **Identify the Time Period of a Second Pendulum:**
A second pendulum has a time period of 2 seconds. Therefore, we can write:
\[ T = 2 \text{ seconds} \]
This means:
\[ 2 = 2\pi \sqrt{\frac{L}{g}} \]

3. **Solve for the Length \( L \) of the Second Pendulum:**
Rearranging the equation to find \( L \):
\[ 1 = \pi \sqrt{\frac{L}{g}} \]
Squaring both sides:
\[ 1 = \pi^2 \frac{L}{g} \]
Thus:
\[ L = \frac{g}{\pi^2} \]

4. **Determine the Length of the New Pendulum:**
The new pendulum has a length that is one third of the second pendulum:
\[ L' = \frac{1}{3}L = \frac{1}{3} \cdot \frac{g}{\pi^2} = \frac{g}{3\pi^2} \]

5. **Calculate the Time Period for the New Pendulum:**
Using the time period formula for the new length \( L' \):
\[ T' = 2\pi \sqrt{\frac{L'}{g}} = 2\pi \sqrt{\frac{\frac{g}{3\pi^2}}{g}} \]
Simplifying:
\[ T' = 2\pi \sqrt{\frac{1}{3\pi^2}} = 2\pi \cdot \frac{1}{\pi \sqrt{3}} = \frac{2}{\sqrt{3}} \text{ seconds} \]

6. **Final Answer:**
The time of oscillation for the pendulum whose length is one third of the second pendulum is:
\[ T' = \frac{2}{\sqrt{3}} \text{ seconds} \]