Energy is conserved for SHM; the sum of the potential and kinetic energies is constant.

The potential energy is: U = ½ k
x^{2} = ½ k A^{2} cos^{2} (ω t)

The kinetic energy is: K =
½ m v^{2} = ½ m ω^{2} A^{2}
sin^{2} (ω t)

The total energy E = K + U is:

½ k A^{2}
cos^{2} (ω t) + ½ m ω^{2} A^{2}
sin^{2} (ω t)

Using k = m ω^{2}, we find

E = ½ k A^{2}
cos^{2} (ω t) + ½ k A^{2} sin^{2} (ω t)
= ½ k A^{2}

Notice that E also equals the maximum potential energy:

U_{max} = ½
kA^{2}

E also equals the maximum kinetic energy:

K_{max} = ½ m
v_{max}^{2} = ½ mA^{2} ω^{2} =
½ k A^{2}

The potential and kinetic energies go through two cycles for each amplitude cycle.

The first set of graphs is for ω = 1 rad/s. The second set is for ω = 0.8 rad/s. This change of ω is accomplished either by decreasing the spring constant or increasing the mass. Which change was made?

- Decreased the spring constant
- Increased the mass
- Either, you can't tell the difference