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A trolley of mass $m_1$ is placed on horizontal rigid pair of rails at same height. A mass $m_2$ is suspended to the trolley vertically by me ans of a ideal massless rope. The rope hangs between rails without touching them. Trolley can move along smooth rails but can't move in any other direction. Suspended mass is given small oscillation and perform $SHM$ after displacing small from stable equilibrium position in two ways, first perpendicular to the rails and second parallel to the rails. The ratio of time period of these (second case to first case) two $SHM's$ is
$\sqrt {\frac{{{m_1}}}{{{m_{`1}} + {m_2}}}} $
$\sqrt {\frac{{{m_2}}}{{{m_{`1}} + {m_2}}}} $
$\sqrt {\frac{{{m_1}}}{{{m_{`2}}}}} $
$\sqrt {\frac{{{m_2}}}{{{m_{`1}}}}} $
Solution
For oscillation perpendicular to plane it is
$\mathrm{T}_{1}=2 \pi \sqrt{\frac{l}{\mathrm{g}}}$ but for case of $\mathrm{SHM}$ in the plane
of rails $\mathrm{T}_{2}=2 \pi \sqrt{\frac{\mathrm{m}_{1} l}{\left(\mathrm{m}_{1}+\mathrm{m}_{2}\right) \mathrm{g}}}$ so $\frac{\mathrm{T}_{2}}{\mathrm{T}_{1}}=\sqrt{\frac{\mathrm{m}_{1}}{\mathrm{m}_{1}+\mathrm{m}_{2}}}$
