Show that the usual formula \(P=2 \pi \sqrt{R / g}\) for the period of small-amplitude oscillations of a pendulum of length \(R\) becomes instead \(P=2 \pi \sqrt{R / g}\left(\sqrt{m_{\mathrm{I}} / m_{\mathrm{G}}}\right)\) if the inertial and gravitational masses of the pendulum bob differ. (Newton himself built pendulums with plumb bobs made of different materials. He would swing two of them side by side, both with the same length \(R\), to see if he could detect a difference in period apart from experimental errors. He could not. Nevertheless, it is interesting that he conceived of the possibility they might be different.)