Question

State work energy theorem. Plot spring force F versus x and obtain the expression for elastic potential energy of spring.

Answer 1

Work-Energy Theorem states Sum of work done by all forces on a body is equal to the change of kinetic energy of the body.

$\sum _i^{}{W}_j=\frac{1}{2}m(v_f^2-v_i^2)$

where $W_j$ is the work done by the $j$ force, and $v_f$ and $v_i$ are final and initial velocities

We know spring force $F=-kx$, where $k$ is called the spring constant.

Hooke law states the spring force $F=-kx$

We consider a string of string constant $k$ and initial elongation $x$ we stretch it by an amount $dx$

So, Work done to stretching The spring gets stared as elastic potential energy in it and is equal to

$dU=dW=Fdx\cos {180}^o$ [$\because$ The force and the resulting displacement are anti-parallel]

Integrating with proper limits

$U_{elastic}={\int }_0^x-kxdx(-1)=\frac{1}{2}k[x^2{]}_0^x$

Or $U_{elastic}=\frac{1}{2}k{x}^2$