## Question

A spring of negligible mass having a force constant *k* extends by an amount *y* when a mass *m* is hung from it. The mass is pulled down a little and released. The system begins to execute simple harmonic motion of amplitude A and angular frequency ω. The total energy of the mass-spring system will be

### Solution

Let L be the relaxed length of the spring and *y* the extension produced in it due to force *mg* so that

*ky* = *mg* …(i)

The displacement of the mass during oscillation is given by

…(ii)

At the instant when the displacement is *x*

Taking gravitational PE at the mean position to be zero,

Gravitational PE at *x* = – mg *x * … (v)

Adding (iii), (iv) and (v), we get

Total energy of mass-spring system

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