Question

By what factor should the length of the pendulum of a clock be changed if you want it to run 2 times faster?

Answer 1

The period of a simple pendulum is given by $T=2\pi \sqrt{ \frac{l}{g} }$.

Here $T$ is the period, $l$ length of the pendulum and $g$ acceleration due to gravity. When the pendulum runs 2 times faster, the period gets halved. That is

$\frac{T}{2}=2\pi \sqrt{ \frac{\frac{l}{4}}{g} }$. The new length of the pendulum is $\frac{l}{4}$.

That is the length of the pendulum decreases by a factor of 4.

Answer 2

You have to decrease the length of the pendulum by 4 times in order to make the clock go 2 times faster