Two spring systems have equal mass and spring constants $k_{1}$ and $k_{2}$ . If the maximum velocities in two systems are equal, then the ratio of amplitude of first to that of second is,

\sqrt{\frac{k_{1}}{k_{2}}}

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\frac{k_{1}}{k_{2}}

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\frac{k_{2}}{k_{1}}

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\sqrt{\frac{k_{2}}{k_{1}}}

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Solution

The correct option is D $\sqrt{\frac{k_{2}}{k_{1}}}$
The Angular frequency of a spring-mass system is, $ω = \sqrt{\frac{k}{m}}$
And the maximum velocity is $v = A ω$
but $v_{1} = v_{2}$
$\Rightarrow A_{1} ω_{1} = A_{2} ω_{2}$
$\Rightarrow A_{1} \sqrt{\frac{k_{1}}{m}} = A_{2} \sqrt{\frac{k_{2}}{m}}$
$\Rightarrow \frac{A_{1}}{A_{2}} = \sqrt{\frac{k_{2}}{k_{1}}}$

Hence, $(D)$ is the correct answer.

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