b
(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\( = \frac{{m'g}}{k}\)

\( = \frac{{13}}{{36}} \times \frac{{mg}}{k}\)\( = \frac{{13}}{{36}} \times 10 \times \frac{{0.36}}{{4{\pi ^2}}} \approx 3.5\;cm\)