A body of mass m is attached to the spring which is elongated to 25 cm by an applied for from its equilibrium position,
(a) Calculate the potential energy stored in the spring-mass system?
(b) What is the work done by the spring force in this elongation?
(C) Suppose the spring is compressed to the same 25 cm, calculate the potential energy stored and also the work done by the spring force during compression. (The spring constant,`k=0.1Nm^(-1))`.

Explanation

The spring constant, `k=0.1Nm^(-1)`
The displacement `x=25cm=0.25m`
(a) The potential energy stored in the spring is given by
`U=1/2Kx^(2)=1/2xx0.1xx(0.25)^(2)=0.0031J`
(b) The work done `W_(s)` by the spring force `vecF_(s)` is given by,
`W_(s)=underset(0)overset(x)vecF_(x)int*dvecr=underset(0)overset(x)int(-kxhati)*(dxhati)`
This spring force `vecF_(s)` acts in the negative x direction while elongation acts in the positive x direction.
`W_(s)=underset(0)overset(x)int(-kx)dx=-1/2kx^(2)`
`w_(s)=-1/2x0.1xx(0.250^(2)=-0.003J`
Note that the potential energy is defined through the work done by the agency The positive sign in the potential energy implies that the energy is reed from the agency to the object. But the work done by the receng for indicais crester is in the opposite direction to the displacement direction.
(c) During compression also the potential energy stored in the object is the same `U=1/2KX^(2)=0.0031J`
Work done by the restoring som force during given by
`W_(s)=underset(0)overset(x)vecF_(x)int*dvecr=underset(0)overset(x)int(kxhati)*(-dxhati)`
In the case of compression, the restoring spring force towards postes and displacement is along negative x direction
`W_(s)=underset(0)overset(x)int(-kx)dx=-1/2kx^(2)=-0.0031J`