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\centerline{\bf Special and General Relativity (PHZ 4601/5606) Fall 2017 
Solutions} \medskip

\centerline{\bf Set 2} 

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% \end{figure} \centerline{Potential as function of $z$.}

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\item{\bf 3. Small oscillations and $m_I/m_G$.} 
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The tangential force of the bob is 
$$ F = m_G\,g\,\sin\theta \approx m_G\,g\,\theta\,,$$
where the approximation defines small oscillations.
By Newton's second law the acceleration is given by
$$ F = - m_I\,l\,\ddot\theta\ \Rightarrow\ \ddot\theta =
   - \frac{m_G\,g}{m_I\,l}\,\theta\,, $$
and with suitable initial conditions the solutions is
$$ \theta = const\,\cos\sqrt{\frac{m_G\,g}{m_I\,l}}\,t $$
with period
$$ T = 2\pi\,\sqrt{\frac{m_I}{m_G}}\,\sqrt{\frac{l}{g}}\,.$$
If the period depends only on $l$ and $g$, then $m_I/m_G$ must 
be the same for all bobs independently of the material used.

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