A mass x gram is suspended from a light spring. It is pulled in downward direction and released so that mass performs S.H.M. of period T. If mass is increased by Y gram, the period becomes 4 T /3 . Th

Question

A mass x gram is suspended from a light spring. It is pulled in downward direction and released so that mass performs S.H.M. of period T. If mass is increased by Y gram, the period becomes 4 T /3 . The ratio of Y / x is

TestHub · Accepted Answer

The time period of a spring-mass system is given by $T = 2 π \frac{m}{k}$ .

For the initial mass $x$ , the period is:
$T = 2 π \frac{x}{k}$

When the mass is increased by $Y$ , the new mass is $x + Y$ and the new period is $T^{'} = \frac{4 T}{3}$ .
$T^{'} = 2 π \frac{x + Y}{k}$

Taking the ratio of the two periods:
$\frac{T ^{'}}{T} = \frac{x + Y}{x}$

$\frac{4}{3} = 1 + \frac{Y}{x}$

Squaring both sides:
$\frac{16}{9} = 1 + \frac{Y}{x}$

$\frac{Y}{x} = \frac{16}{9} - 1 = \frac{7}{9}$

The ratio of $Y / x$ is $7 : 9$ .

Answer: $7 : 9$

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