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A body of mass $m$ is attached to one end of a massless spring which is suspended vertically from a fixed point. The mass is held in hand, so that the spring is neither stretched nor compressed. Suddenly the support of the hand is removed. The lowest position attained by the mass during oscillation is $4\,cm$ below the point, where it was held in hand.
$(a)$ What is the amplitude of oscillation ?
$(b)$ Find the frequency of oscillation.
Solution
$(a)$ When support of the hand is removed, the body of mass $m$ oscillate about mean position.
Suppose, a body reaches at lower extreme point and the maximum extension of spring is $x$,
$\therefore$ The decrease in potential energy of body $=m g x$
Increase in elastic potential energy of spring $=\frac{1}{2} k x^{2}$
Now mechanical energy is conserved
$\therefore m g x=\frac{1}{2} k x^{2}$
$\therefore x=\frac{2 m g}{k} \quad \ldots \text { (1) }$
