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We that know how **potential energy** is defined for any **conservative forces**, here we will derive the Potential energy of **spring**. All the results are derived assuming **ideal springs**.

Inside Story

# The potential energy of Spring

### Meaning of potential energy of spring

We can see that if we compress or elongate a **spring**, we feel opposing force by spring, Whenever we disturb spring from the natural length it tries to retain its natural length, therefore there is some energy stored in a compressed or stretched spring due to which it tries to retain natural length.

Actually, this energy stored in any stretched/compressed spring or in stretching of any rubber band, etc, is elastic potential energy as the bodies are resisting change in its shape, size, length, etc.

### Derivation of the potential energy of spring

#### When compressing a spring from the natural length

**Spring force** is conservative therefore by **conservative field theory** we know that

(1)

Suppose we have a spring of natural length “l”. We compress it by x length from the natural length, now in this compressed state spring is capable of doing **work** hence having energy.

Now **work done** by spring in compressing

Because force and displacement are opposite.

(2)

From equation (1) and (2)

If we take x = 0 i.e. natural length and take it as a reference point and assume the **potential energy** at natural length to be zero, then

(3)

compress a spring by an amount “x” w.r.t natural length the potential energy stored in spring at that state is given by equation (3).

#### When stretching a spring from natural length.

We know that whether we stretch or compress a spring by “x” amount from the natural length the **work done by spring in both cases is the same.**

Therefore if we strech spring by “x” amount.

(4)

From equation (1) and (4) .

If we take x = 0 i.e. natural length and take it as a reference point and assume the **potential energy** at natural length to be zero, then

(5)

Stretch a spring by an amount “x” w.r.t natural length the potential energy stored in spring at that state is given by equation (5).

# Result

From equations (3) and (5), we can see that whether we compress or stretch a spring by an amount “x” the amount of potential energy of spring in that state is the same in both springs.

Given by

(6)

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