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A pan with set of weights is attached with a light spring. When disturbed, the mass-spring system oscillates with a time period of $0.6$ $s.$ When some additional weights are added then time period is $0.7s.$ The extension caused by the additional weights is approximately given by ......... $cm$
$1.38$
$3.5 $
$1.75$
$2.45$
Solution
(b) $2\pi \sqrt {\frac{m}{k}} = 0.6$ …$(i)$ and $2\pi \sqrt {\frac{{m + m'}}{k}} = 0.7$ …$(ii)$
Dividing $(ii)$ by $(i)$ we get ${\left( {\frac{7}{6}} \right)^2} = \frac{{m + m'}}{m} = \frac{{49}}{{36}}$
$\frac{{m + m'}}{m} – 1 = \frac{{49}}{{36}} – 1 \Rightarrow \frac{{m'}}{m} = \frac{{13}}{{36}}$ $⇒$ $m' = \frac{{13m}}{{36}}$
Also $\frac{k}{m} = \frac{{4{\pi ^2}}}{{{{(0.6)}^2}}}$
Desired extension $ = \frac{{m'g}}{k}$$ = \frac{{13}}{{36}} \times \frac{{mg}}{k}$
$ = \frac{{13}}{{36}} \times 10 \times \frac{{0.36}}{{4{\pi ^2}}} \approx 3.5\;cm$
