By what factor should the length of a simple pendulum be changed if the period of vibration were to be increased by a factor of three? A)1/3² B)1/3 C)3 D)3² E)1

Question

Quizplus · Accepted Answer

The period of a simple pendulum is given by $T = 2\pi\sqrt{\frac{L}{g}}$, where $T$ is the period, $L$ is the length of the pendulum, and $g$ is the acceleration due to gravity. If the period is to be increased by a factor of three, the equation becomes $3T = 2\pi\sqrt{\frac{L'}{g}}$, where $L'$ is the new length. Squaring both sides of the equation to eliminate the square root gives $(3T)^2 = (2\pi)^2\frac{L'}{g}$, and since $T^2 = (2\pi)^2\frac{L}{g}$, it follows that $9T^2 = (2\pi)^2\frac{L'}{g}$. Therefore, $L'$ must be 9 times $L$, which means the length must be changed by a factor of $3^2$ or $9$, corresponding to $3^2$.

By What Factor Should the Length of a Simple Pendulum