If a Seconds pendulum is moved to a planet where acceleration due to gravity is 4 times, the length of the second's pendulum on the planet should be made

Options

A. 2 times
B. 4 times
C. 8 times
D. 15 times

Answer

To solve the problem of determining the length of a seconds pendulum on a planet where the acceleration due to gravity is four times that on Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of a Seconds Pendulum**:
A seconds pendulum is defined as a pendulum that has a time period of 2 seconds. 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. **Set Up the Equation for Earth**:
For a seconds pendulum on Earth, we have:
\[ T = 2 \text{ seconds} = 2\pi \sqrt{\frac{L}{g}} \]
Rearranging this gives us:
\[ \sqrt{\frac{L}{g}} = \frac{1}{\pi} \]
Squaring both sides results in:
\[ \frac{L}{g} = \frac{1}{\pi^2} \]
Thus, we can express the length \( L \) as:
\[ L = \frac{g}{\pi^2} \]

3. **Consider the New Planet**:
On the new planet, the acceleration due to gravity \( g' \) is four times that of Earth:
\[ g' = 4g \]

4. **Set Up the Equation for the New Planet**:
The time period on the new planet must still be 2 seconds, so we have:
\[ T = 2\pi \sqrt{\frac{L'}{g'}} \]
where \( L' \) is the new length of the pendulum. Rearranging gives:
\[ \sqrt{\frac{L'}{g'}} = \frac{1}{\pi} \]
Squaring both sides results in:
\[ \frac{L'}{g'} = \frac{1}{\pi^2} \]
Thus, we can express the new length \( L' \) as:
\[ L' = \frac{g'}{\pi^2} \]

5. **Substituting the New Gravity**:
Now substitute \( g' = 4g \) into the equation for \( L' \):
\[ L' = \frac{4g}{\pi^2} \]

6. **Relate the New Length to the Original Length**:
From our earlier expression for \( L \):
\[ L = \frac{g}{\pi^2} \]
Therefore, we can relate \( L' \) to \( L \):
\[ L' = 4 \cdot \frac{g}{\pi^2} = 4L \]

### Conclusion:
The length of the seconds pendulum on the new planet should be **4 times the original length**.