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(A) The time period of a mass-spring system is given by $T = 2\pi \sqrt{\frac{m}{k}}$, where $m$ is the mass and $k$ is the spring constant.For the fi

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$4 : 5$

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(A) The time period of a mass-spring system is given by $T = 2\pi \sqrt{\frac{m}{k}}$, where $m$ is the mass and $k$ is the spring constant.For the first case, $T = 2\pi \sqrt{\frac{m_1}{k}}$.For the second case, the total mass is $(m_1 + m_2)$, so the new time period is $T' = 2\pi \sqrt{\frac{m_1 + m_2}{k}} = \frac{3}{2} T$.Dividing the two equations: $\frac{T'}{T} = \sqrt{\frac{m_1 + m_2}{m_1}} = \frac{3}{2}$.Squaring both sides: $\frac{m_1 + m_2}{m_1} = \frac{9}{4}$.This simplifies to $1 + \frac{m_2}{m_1} = \frac{9}{4}$, which means $\frac{m_2}{m_1} = \frac{9}{4} - 1 = \frac{5}{4}$.Therefore, the ratio $\frac{m_1}{m_2} = \frac{4}{5}$.

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