# How did Newton prove the equivalence principle with pendulums?

I know pendulums may be equivalent to a free fall but I would appreciate any further explanation for it.

• What is it exactly that you're asking? – gented Jul 12 '17 at 15:47

I think that the equivalence principle you are talking about is the one that states "inertial mass is equivalent to gravitational mass".

When you have a pendulum, the restoring force is proportional to the force of gravity, and the $\sin$ of the angle between the pendulum and the vertical. The inertial force (the mass resisting acceleration) depends just on the (mass of the) material.

The equation of motion is given by

$$m_i \ell \ddot \theta = -m_g g \sin\theta$$

Where I deliberately leave the "inertial mass" $m_i$ and the "gravitational mass" $m_g$ as two different terms.

For a small angle of deflection, where we can take $\sin\theta \approx \theta$, this equation can be solved. It gives us for the period

$$T = 2\pi \sqrt\frac{m_i \ell}{m_g g}$$

Normally, we take it that $m_i = m_g$, in which case the expression reduces to the familiar one.

If there is non-equivalence, then you would expect this to show up when comparing different materials (say lead and silver) - if they have different ratios for $m_i/m_g$, then an otherwise identically constructed pendulum with either lead or silver as the bob would have a different period (controlling for things like drag - which has a very small effect for a heavy bob on a long string with a small deviation; to eliminate the effect of difference in drag, Newton put the different masses in identical wooden boxes so they had the exact same drag).

Set up two pendulums side by side, and let them go at the same time. Observe whether they stay "in sync", or whether one is swinging faster than the other. Repeat with different materials.

I found a description of the experiments he actually did:

Newton's profound insight was that this Equivalence was essential to understanding the laws of motion, and he devised two tests of it. First, he made two identical pendulums, each 11 feet long ending in a wooden box. One was a reference; he started the pendulums together and timed how quickly their cycles drifted apart. In the other he put successively "gold, silver, lead, glass, common salt, wood, water and wheat". If Equivalence holds, the times of swing should be independent of the material, and they were -- to better than a part in 1000. Good enough for government work, and good enough for Newton. Second, Newton realized that if Equivalence is wrong, the motions of Jupiter's moons and of the Earth-Moon system in the gravity of the Sun would be greatly altered. After further refinement by Pierre-Simon Laplace in 1787, this argument yielded a test of Equivalence to a few parts in 10$^7$.

And the original description (from Principia, book III, proposition VI):

I found this on this site which reproduced this 1846 translation by Andrew Motte).

Imagine a pendulum for which the ball at the end of the string (of length $l$) would have an inertial mass $m_I$ different from its gravitational mass $m_G$. Then the equation of motion would be

$$m_I l \ddot{\theta}=-m_Gg\sin\theta,$$

i.e. in the approximation of small oscillations

$$\ddot{\theta} + \frac{m_G}{m_I}\frac{g}{l}\theta=0,$$

which results in harmonic motion with a period

$$T=T_0 \sqrt{\frac{m_I}{m_G}},$$

where $T_0$ would be the period if the equivalence principle were true,

$$T_0=2\pi\sqrt{\frac{l}{g}}.$$

So now, we make several pendulums using different materials for the ball and we swing them together. If $m_I/m_G$ varies with the material, the pendulums will get out of synch. Newton did such an experiment and could not measure any difference between the periods of the different pendulums. As a result, he concluded that the ratio $m_I/m_G$ is the same for all materials. I do not know what precision Newton could reach.