Question

problem 6: traditional pendulum clocks are made so the pendulum takes 2 seconds to do one complete cycle (2 swings per cycle). approximate the length of the pendulum arm, given that most of the weight is at the lower end.

Explanation:

Step1: Recall the formula for the period of a simple pendulum

The formula for the period \(T\) of a simple pendulum is \(T = 2\pi\sqrt{\frac{L}{g}}\), where \(T\) is the period, \(L\) is the length of the pendulum, and \(g\approx9.8\space m/s^{2}\). Given \(T = 2\space s\).

Step2: Solve the formula for \(L\)

First, square both sides of the equation \(T = 2\pi\sqrt{\frac{L}{g}}\) to get \(T^{2}=4\pi^{2}\frac{L}{g}\). Then, solve for \(L\): \(L=\frac{T^{2}g}{4\pi^{2}}\).

Step3: Substitute the values of \(T\) and \(g\)

Substitute \(T = 2\space s\) and \(g = 9.8\space m/s^{2}\) into the formula \(L=\frac{T^{2}g}{4\pi^{2}}\). So \(L=\frac{2^{2}\times9.8}{4\pi^{2}}=\frac{4\times9.8}{4\pi^{2}}=\frac{9.8}{\pi^{2}}\).

Step4: Calculate the value of \(L\)

Since \(\pi\approx3.14\), then \(\pi^{2}\approx9.86\). So \(L=\frac{9.8}{9.86}\approx1\space m\).

Answer:

The length of the pendulum arm is approximately \(1\space m\).