When a mass $m$ is attached to a spring it oscillates with period $4 \,s$. When an additional mass of $2 \,kg$ is attached to a spring, time period increases by $1 \,s$. The value of $m$ is ........... $kg$
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(b)
$\omega_1=\sqrt{\frac{k}{m}}$
$\omega_2=\sqrt{\frac{k}{m+2}}$
$\frac{\omega_1}{\omega_2}=\sqrt{\frac{m+2}{m}}$
Since $\omega=\frac{2 \pi}{T}$
$\frac{T_2}{T_1}=\sqrt{\frac{m+2}{m}}$
$\left(\frac{5}{4}\right)^2=\frac{m+2}{m}$
$\frac{25}{16}=\frac{m+2}{m}$
$25 \,m=16 \,m+32$
$9 \,m=32$
$m=3 \frac{5}{9} \,kg$
$m=3.5 \,kg$
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