Energy

The spring is a conservative force. Thus:

$$U = -W(0\to x) = - \int_0^x (-k x)dx = \frac{1}{2} k x^2$$ (9.17)

$$= \frac{1}{2} k X_0^2 \cos^2(\omega t + \phi)$$ (9.18)

where we have arbitrarily set the zero of potential energy to be the equilibrium position (what would it look like if the zero were at $x_0$?).

The kinetic energy is:

$$K = \frac{1}{2} m v^2$$ (9.19)

$$= \frac{1}{2} m (\omega^2) X_0^2 \sin^2(\omega t + \phi)$$ (9.20)

$$= \frac{1}{2} m (\frac{k}{m}) X_0^2 \sin^2(\omega t + \phi)$$ (9.21)

$$= \frac{1}{2} k X_0^2 \sin^2(\omega t + \phi)$$ (9.22)

The total energy is thus:

$$E = \frac{1}{2} k X_0^2 \sin^2(\omega t + \phi) + \frac{1}{2} k X_0^2 \cos^2(\omega t + \phi)$$ (9.23)

$$= \frac{1}{2} k X_0^2$$ (9.24)

and is constant in time! Kinda spooky how that works out...

Note that the energy oscillates between being all potential at the extreme ends of the swing (where the object comes to rest) and all kinetic at the equilibrium position (where the object experiences no force).