Physics Problem: Rearranging Formulas

1. Key Definitions and Formulas

Before solving the problem, it is essential to understand the variables involved in the formula for a simple pendulum:

Period (\(T\)): The time required for one complete cycle of oscillation (back and forth). It is typically measured in seconds (\(s\)).
Length (\(L\)): The distance from the pivot point to the center of mass of the pendulum bob. It is typically measured in meters (\(m\)).
Acceleration due to Gravity (\(g\)): The constant acceleration exerted by gravity on a free-falling object. On Earth, this is approximately \(9.81 \, m/s^2\).
The Pendulum Formula: \(\(T = 2\pi \sqrt{\frac{L}{g}}\)\)

Task: Rearrange the equation \(T = 2\pi \sqrt{\frac{L}{g}}\) to solve for \(g\).

Start with the original formula: \(\(T = 2\pi \sqrt{\frac{L}{g}}\)\)
Divide both sides by \(2\pi\) to isolate the square root: \(\(\frac{T}{2\pi} = \sqrt{\frac{L}{g}}\)\)
Square both sides of the equation to remove the square root: \(\(\left(\frac{T}{2\pi}\right)^2 = \frac{L}{g}\)\)
Distribute the square to the numerator and denominator: \(\(\frac{T^2}{4\pi^2} = \frac{L}{g}\)\)
Multiply both sides by \(g\) to move it out of the denominator: \(\(g \cdot \frac{T^2}{4\pi^2} = L\)\)
Multiply both sides by \(\frac{4\pi^2}{T^2}\) (or divide by \(\frac{T^2}{4\pi^2}\)) to isolate \(g\): \(\(g = \frac{4\pi^2 L}{T^2}\)\)

The rearranged formula for the acceleration due to gravity is: \(\(g = \frac{4\pi^2 L}{T^2}\)\)