A mass ' \(m\) ' attached to a spring oscillates with a period of 3 second. If the mass is increased by 0.6 kg , the period increases by 3 second. The initial mass ' m ' is equal to

Time period,
\(\begin{array}{rlrl}
& & \mathrm{T} & =2 \pi \sqrt{\frac{\mathrm{~m}}{\mathrm{k}}} \\
& \therefore \quad \mathrm{~T}_1 & =2 \pi \sqrt{\frac{\mathrm{~m}}{\mathrm{k}}} \\
& \therefore \quad 3 & =2 \pi \sqrt{\frac{\mathrm{~m}}{\mathrm{k}}} \\
& \frac{9}{4 \pi^2} & =\frac{\mathrm{m}}{\mathrm{k}} \\
& \therefore \quad \mathrm{k} & =\frac{4 \pi^2 \mathrm{~m}}{9}
\end{array}\)
...(given, \(\mathrm{T}_1=3 \mathrm{~s}\) )
When mass is increased by 0.6 kg ,
\(\mathrm{T}_2=2 \pi \sqrt{\frac{\mathrm{~m}+0.6}{\mathrm{k}}}\)
...[From(i)]
\(6=\sqrt{\frac{m+0.6}{k}}\).
... (given \(\mathrm{T}_2=3+3=6 \mathrm{~s}\) )
\(\frac{36}{4 \pi^2}=\frac{m+0.6}{k}\)
\(\begin{aligned} & \frac{36}{4 \pi^2}=(\mathrm{m}+0.6) \frac{9}{4 \pi^2 \mathrm{~m}} \\ & 4 \mathrm{~m}=\mathrm{m}+0.6 \\ & \mathrm{~m}=0.2 \mathrm{~kg}\end{aligned}\)