03 Physical Pendulum (3) Oscillating Ring

Aim¶

To show a particular example of a compound pendulum.

Subjects¶

• 3A15 (Physical Pendula)

Diagram¶

Equipment¶

• 2 large (steel) rings, $\varphi=600$ with knife-edge suspension. Th ese rings can be divided into $2 / 3$ and $1 / 3$.

• mathematical pendulum, $\mathrm{I}=600$

• meterstick

Presentation¶

One complete ring swings in its plane at the knife-edge on its periphery. A simple pendulum whose length is equal to the diameter of the ring is suspended beside it so the equality of periods can be observed.

A second $2 / 3$-ring is made swinging. It can be observed that the ring has still the same period!

Again the same period is measured when $1 / 3$-ring is swinging

Explanation¶

• For a physical pendulum, the period $T$ is given by $T=\frac{2 \pi}{\sqrt{g}} \sqrt{\frac{I_{A}}{m s}}$.

  If the pendulum is a complete ring, then $s=R$ (see Figure 2), $I_{A}=I_{C}+m R^{2}$ and $I_{c}=m R^{2}$. Then $T=\frac{2 \pi}{\sqrt{g}} \sqrt{2 R}$, so $I_{r}=2 R$.

So a complete ring has the same period as a mathematical pendulum of length 2R.

• If the pendulum is part of a complete ring, $I_{O}=m R^{2}$ (Figure 3). Also $I_{O}=I_{C}+m(R-$ $s)^{2}$ (C is the center of mass) and $I_{A}=I_{C}+m s^{2}$. It follows that $I_{A}=2 m R s$ and $T=\frac{2 \pi}{\sqrt{g}} \sqrt{2 R}$. So again $I_{r}=2 R$.

Sources¶

• Ehrlich, R., Why Toast Lands Jelly-Side Down: Zen and the Art of Physics Demonstrations, pag. 126-127

• Roest, R., Inleiding Mechanica, pag. 169-170

• Sutton, Richard Manliffe, Demonstration experiments in Physics, pag. 88