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Operations on Rational and Irrational Numbers

A body is in ...

Question

A body is in $S H M$ with period $T$ when oscillated from a freely suspended spring. If this spring is cut in two parts of length ratio $1 : 3$ & again oscillated from the two parts separately, then the periods are $T_{1}$ & $T_{2}$ then find $T_{1} / T_{2}$

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Solution

R.E.F image
$F = - k l$
$a = - \frac{k}{m} l$
$\Rightarrow w = \sqrt{\frac{k}{m}}$
$\Rightarrow T = 2 π \sqrt{\frac{m}{k}}$
for $l / 4 : -$
$F = \frac{- k}{4} l$
$\Rightarrow a = \frac{- k}{4 m} l$
$\Rightarrow w = \sqrt{\frac{k}{4 m}} l$
$\Rightarrow w = \sqrt{\frac{k}{4 m}}$
$\Rightarrow T_{1} = 2 π \sqrt{\frac{4 m}{k}}$
for $3 l / 4 : -$
$F = \frac{- 3 k}{4} l$
$\Rightarrow a = - \frac{3 k}{4 m} l$
$\Rightarrow w = \sqrt{\frac{3 k}{4 m}}$
$\Rightarrow T_{2} = 2 π \sqrt{\frac{4 m}{3 k}}$
So, $\frac{T_{1}}{T_{2}} = \frac{2 π \sqrt{\frac{4 m}{k}}}{2 π \sqrt{\frac{4 m}{3 k}}}$
$\Rightarrow \frac{T_{1}}{T_{2}} = \frac{\sqrt{3}}{1}$

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