Energy in SHM

We can define simple harmonic oscillation by the equation

(1)

The work done by the force F during a displacement from x to x + dx is

(2)

Work done in a displacement from 0 to x is

(3)

Let U(x) be the potential energy of the system when the displacement is x.

As the change in potential energy corresponding to a force is negative of the work done by this force,

U(x) - U(0) = - W = (4)

We know that Potential energy at x=0 is 0

Hence,

U(x) = 0 and U(x) =

We know that spring constant

Hence we can write the above equation,

U(x) =

The displacement of the particle is

(5)

The velocity of the particle is

(6)

Substituting equation (5) in potential energy expression (4)

U(x) = (7)

Kinetic energy at time t is,

(8)

Substituting equation (6) in equation (8)

Hence the total energy E will be

E = U(x)+K.E

E = +

E = (note: = 1)

Hence,

E =

Hence the total energy at time t is independent of t.

Thus the total mechanical energy remains constant.