The Potential Energy of Spring

HOOKS LAW

The restoring force developed in the spring is proportional to the displacement x and it os opposite to the displacement

ie,

Fα -x

F=-kx

Where k is a constant called spring constant.

Potential energy stored in spring

Consider a massless spring fixed to a rigid support at one end and body attached to the other end. The body moves on frictionless surface.

If a body is displaced by a distance dx, The workdone for this displacement

dw=Fdx

The total workdone to move the body from x=0 to x.

This workdone is stored a potential energy in a spring.

P. E.= 1/2 (kx²)

Spring force is conservative force

If the spring is displaced from initial position xi to x f and again to xi;

Total workdone =

W=0

This zero workdone means, spring force is conservative

Energy of oscillating spring at any point

. If the block of mass ( attached to massless spring)is extended to xₘ and released, it will oscillate in between +xₘ and -xₘ. The total mechanical energy at any point x (lies between -xₘ and +xₘ) is

1/2Kx²ₘ= 1/2(Kx²)+1/2v²

This block mass m has maximum velocity at equilibrium position (x=0) At this position, the potential energy stored in a spring is completely converted in to kinetic energy. ie;

Graphical variation of energy

Variation of Spring force with distance

The area of triangle OBA represents the workdone by the spring force. Due to the opposing signs of Fₓ, this work done is negative

Wₛ=-(1/2)kx²ₘ

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