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
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=-k∫x0xdx=-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]