Question

(a) Find an expression for the work done by the spring force and hence show that the spring force is a conservative force. Also deduce an expression for the potential energy of an stretched spring​

Answer 1

Answer:

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Explanation:

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Answer 2

The derivation is given below:

We know that if the displacement of the spring (extension or contraction) is x and the spring constant is k

Then the spring force is

$F=-kx$

if initially the extension (or contraction) is $x_i$ and final extension (or compression) is $x_f$ then the work done

$W=\int Fdx$

$\implies W=\int_{x_i}^{x_f}(-kx)dx$

$\implies W=-\int_{x_i}^{x_f} kxdx$

$\implies W=-\frac{1}{2}k(x_f-x_i)^2$   ..... (1)

If $x_i=0$ and $x_f=x$

Then

$W=-\frac{1}{2}kx^2$

This is the expression for the work done by the spring force

Work done by an external force

$W=\frac{1}{2}kx^2$

A force is conservative when the work done depends upon only the initial and final positions and not on the path.

From (1) it is clear that the work done only depends upon the initial and final state of the spring, therefore spring force is a conservative force.

The change in potential energy will be equal to the work done by the spring force

Thus,

$\Delta PE=W$

$\implies PE_f-0=\frac{1}{2}kx^2$

$\implies PE_f=\frac{1}{2}kx^2$

Hope this answer is helpful.

Know More:

Q: Derive the expression for the potential energy stored in a spring.

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