# How should we measure the length of a pendulum?

To find the length of the pendulum, would you consider the length from the hinge to the beginning of the sphere or end of the sphere or the centre?

• From the point of support to the centre of the sphere. Jun 1, 2022 at 13:15

The equivalent length is not simply the distance from the hinge to the center of mass. Instead, the equivalent length is $$L=\frac{I}{mr}$$ where $$I$$ is the moment of inertia of the pendulum about the pivot, $$m$$ is the mass of the pendulum, and $$r$$ is the distance from the pivot to the center of mass of the pendulum.
• Note that this correction can be pretty small in some circumstances. For the OP's situation (a sphere of radius $a$ whose center is a distance $r$ from the pivot), the fractional difference can be shown to be $\frac25 a^2/r^2$. For a 1-cm ball bearing at the end of a 1-m string, this will be a difference of one part in $10^4$ — which may well be negligible for a home experiment. Jun 1, 2022 at 14:14
• But for a sphere of radius $a$ with its centre a distance $r$ from the pivot, the moment of inertia about the pivot is$$I=mr^2+m\tfrac 25 a^2$$ If $r$=1.00 m and $a$=1.00 cm, $I= 1.00004\ mr^2$, so for most purposes we needn't worry about the $m\tfrac 25 a^2$ term. Note to Michael Seifert: How amusing! I wrote this comment before I'd seen yours! Jun 1, 2022 at 15:37