A mass $M$ is suspended from a spring of negligible mass. The spring is pulled a little and then released so that the mass executes simple harmonic oscillations with a time period $T$. If the mass is increased by m then the time period becomes $\left( {\frac{5}{4}T} \right)$. The ratio of $\frac{m}{{M}}$ is
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(a) $T = 2\pi \sqrt {\frac{m}{K}} $
==> $m \propto {T^2}$
==> $\frac{{{m_2}}}{{{m_1}}} = \frac{{T_2^2}}{{T_1^2}}$
$ \Rightarrow \frac{{M + m}}{M} = {\left( {\frac{{\frac{5}{4}T}}{T}} \right)^2}$
$ \Rightarrow \frac{m}{M} = \frac{9}{{16}}$
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