Question

A massless spring is hung vertically from a rigid support. A man ' 'm' ' is attached at its free end. The mass is slightly depressed and then released. Show that the motion is simple harmonic and find its time period.​

Answer 1

Given:

A massless spring is hung vertically from a rigid support. A man ' 'm' ' is attached at its free end. The mass is slightly depressed and then released.

To derive:

The motion is simple harmonic and find its time period.

Derivation:

In equilibrium condition , the weight of the man is equal to the spring force:

$\rm{ \therefore \: mg = kx_{0}}$

Now , when the spring is depressed by a small displacement (x) let the net force be F ;

$\rm{ \therefore \: F = k(x_{0} - x) - mg}$

$\rm{ = > \: F = k(x_{0} - x) - kx_{0}}$

$\rm{ = > \: F = kx_{0} - kx - kx_{0}}$

$\rm{ = > \: F = - kx }$

$\rm{ = > \: F \propto - x }$

Hence , the motion is SHM.

Let time period be t ;

$\boxed{ \rm{ \therefore \: t = 2\pi \sqrt{ \dfrac{m}{k} } }}$

Hope It Helps.