A simple pendulum, made of a string of length l and a bob of mass m, is released from a small angle ${\mathrm{\theta }}_0$ . It strikes a block of mass M, kept on a horizontal surface at its lowest point of oscillations, elastically. It bounces back and goes up to an angle ${\mathrm{\theta }}_1$ . Then M is given by :

The velocity of m before collision is $\mathrm{u}=\sqrt{2\mathrm{gl}\left(1-{\mathrm{cos\theta }}_{\mathrm{o}}\right)}$

and the velocity after collision is $\mathrm{v}=\sqrt{2\mathrm{gl}\left(1-{\mathrm{cos\theta }}_1\right)}$.

As the collision is elastic the final velocity after collision is given by $\mathrm{v}=\frac{\mathrm{m}-\mathrm{M}}{\mathrm{m}+\mathrm{M}}\mathrm{u}$

On equating these two, we get $\mathrm{v}=\sqrt{2\mathrm{gl}\left(1-{\mathrm{cos\theta }}_1\right)}=\frac{\mathrm{m}-\mathrm{M}}{\mathrm{m}+\mathrm{M}}\sqrt{2\mathrm{gl}\left(1-{\mathrm{cos\theta }}_0\right)}$

$\frac{\mathrm{m}-\mathrm{M}}{\mathrm{m}+\mathrm{M}}=\frac{1-{\mathrm{cos\theta }}_1}{1-{\mathrm{cos\theta }}_0}=\frac{\sin \frac{{\mathrm{\theta }}_1}{2}}{\sin \frac{{\mathrm{\theta }}_0}{2}}$

As the angle are small we can consider the approximation of $\mathrm{sin\theta }\simeq \mathrm{\theta }$

So $\frac{\mathrm{m}-\mathrm{M}}{\mathrm{m}+\mathrm{M}}=\frac{{\mathrm{\theta }}_1}{{\mathrm{\theta }}_0}$

on solving we get $\frac{\mathrm{M}}{\mathrm{m}}=\frac{{\mathrm{\theta }}_0-{\mathrm{\theta }}_1}{{\mathrm{\theta }}_0+{\mathrm{\theta }}_1}\Rightarrow \mathrm{M}=\left(\frac{{\mathrm{\theta }}_0-{\mathrm{\theta }}_1}{{\mathrm{\theta }}_0+{\mathrm{\theta }}_1}\right)\mathrm{m}$