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MrJazSohani SharmagndurcsathialaAhmedabad

Expression of Potential Energy stored in a Spring

 Statement: Derive the expression of Potential Energy stored in a spring, U=12kx2

Solution:

Elastic potential energy (U) is potential energy stored as a result of deformation of an elastic object, such as the stretching of a spring. It is equal to the work done (W) to stretch the spring, which depends upon the spring constant k as well as the distance stretched i.e. U = W

Fig - 1: A spring system

Let the spring be stretched trough a small distance dx
Then work done in stretching the spring through a distance dx is, 

dW=Fdx ...................................... (1) 

Where, F is the force applied to the stretch the spring.

Total work done in stretching the spring from the interval x=0 to x=x is obtained by integrating the expression:

dW=x0Fdx..............................(2)

Hooke’s Law states that, the elongation produced in an ideal spring is directly proportional to the spring force. That is,

F=-kx ......................................(3) 

 Here, k is called the spring constant.

Substituting equation (iii) in (ii) we get,

Work done by spring force, 

W=x0-kxdx=-kx0xdx=-k[x22]x0=-12kx2 ............................ (4) 

And the work done by external force = 12kx2

This work done to deform the spring is nothing but the elastic potential energy of the spring.

Hence, U=12kx2

[Derived/Proved]

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