I've done enough research before asking this question. The work done by a spring is defined as $$W_\mathrm{spring}=\left|\frac{1}{2}kx^2\right|$$ Where $ k$ is the spring constant and $x$ is the distance moved by the spring. But when I tried to derive the eqñ I'm getting $|kx^2|$ and no half is present. I took into consideration that there is displacement at both ends of the spring even when the force is exerted at only one end but still I end up at this eqñ. Can anyone help me out?
Answer
Say $W$ is:
$$W=\frac{1}{2}kx^2$$
Then:
$$\frac{dW}{dx}=\frac{1}{2} k \times (x^2)'=\frac{1}{2} k \times 2x=kx$$.
But is it was restoring force $F$ you were looking for, then:
$$F=-\frac{dW}{dx}=-kx$$
Inversely:
$$W=\int Fdx=\int(-kx)dx=-k\int xdx= -k\frac{1}{2}x^2=-\frac{1}{2}kx^2$$
(If integrated between the correct boundaries). The sign is a matter of convention.
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