03 Physical Pendulum (3) Oscillating Ring

03 Physical Pendulum (3) Oscillating Ring#

Aim#

To show a particular example of a compound pendulum.

Subjects#

  • 3A15 (Physical Pendula)

Diagram#

../../../../../_images/figure_0129.png

Fig. 305 .#

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 306), \(I_{A}=I_{C}+m R^{2}\) and \(I_{c}=m R^{2}\). Then \(T=\frac{2 \pi}{\sqrt{g}} \sqrt{2 R}\), so \(I_{r}=2 R\).

../../../../../_images/figure_199.png

Fig. 306 .#

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

../../../../../_images/figure_248.png

Fig. 307 .#

  • If the pendulum is part of a complete ring, \(I_{O}=m R^{2}\) (Figure 307). 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

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