`T=2pi sqrt((L)/(g ))`
The formula above was created by Italian scientist Galileo Galilei in the early 1600s to demonstrate that the time it takes for a pendulum to complete a swing-called its period, T-can be found using only the length of the pendulum, L, and the force of gravity, g He proved that the mass of the pendulum did not affect its period. Based on the equation above, which of the following equations could be used to find the length of the pendulum given its period ?

Options

A. `L = (gT)/(2pi)`
B. `L = (g T ^(2))/(4pi^(2))`
C. `L = (T^(2))/(4pi^(2)g)`
D. `L = (g)/(4pi^(2)T ^(2))`

Answer

To find the length of the pendulum \( L \) in terms of the period \( T \) and the force of gravity \( g \) from the equation \( T = 2\pi \sqrt{\frac{L}{g}} \), we can follow these steps:

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]

2. **Square both sides to eliminate the square root**:
\[ T^2 = (2\pi)^2 \cdot \frac{L}{g} \]
This simplifies to:
\[ T^2 = 4\pi^2 \cdot \frac{L}{g} \]

3. **Rearrange the equation to isolate \( L \)**:
Multiply both sides by \( g \):
\[ g \cdot T^2 = 4\pi^2 L \]

4. **Divide both sides by \( 4\pi^2 \)** to solve for \( L \):
\[ L = \frac{g \cdot T^2}{4\pi^2} \]

### Final Result:
Thus, the length of the pendulum \( L \) in terms of the period \( T \) and the force of gravity \( g \) is:
\[ L = \frac{g \cdot T^2}{4\pi^2} \]

### Conclusion:
Based on the options provided, the correct equation to find the length of the pendulum given its period is:
**Option B: \( L = \frac{g \cdot T^2}{4\pi^2} \)**.

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