The question involves a pendulum's period formula:

$P = 2\pi \sqrt{\frac{L}{32}}$

where:

• $P$ is the period (time in seconds for the pendulum to swing back and forth),
• $L$ is the pendulum's length,
• $32$ represents the gravitational constant factor in this context.

The task is to explain how to solve for $L$ given a value of $P$, and identify the correct sequence of algebraic steps among the provided options.

To isolate $L$, you would follow these steps:

1. Square both sides to eliminate the square root: $P^2 = (2\pi)^2 \frac{L}{32}$
2. Divide both sides by $(2\pi)^2$ to isolate $\frac{L}{32}$: $\frac{P^2}{(2\pi)^2} = \frac{L}{32}$
3. Multiply both sides by 32 to solve for $L$: $L = \frac{32 \cdot P^2}{(2\pi)^2}$

Based on these steps, the correct answer should be one that follows this pattern: squaring both sides first, then dividing by $(2\pi)^2$, and finally multiplying by 32.

Would you like more details, or do you have any specific questions?

Related Questions

1. What is the formula for the period of a simple pendulum and what factors affect it?
2. Why is the gravitational constant represented as 32 in this equation?
3. How does changing the length $L$ of a pendulum affect its period $P$?
4. What is the significance of squaring both sides in equations with square roots?
5. How can you rearrange other physics formulas to isolate specific variables?

Tip

When isolating variables in square-root equations, squaring both sides is usually the first step to eliminate the root, making it easier to work with the remaining terms.