Energy in oscillators

(a)

From the general solution

$$x(t)=x_{\mathrm{eq}}+\Re \left( \underline{A} e^{i \omega_o t} \right)$$

determine the total (potential plus kinetic) energy of an undamped oscillator in terms of $\left\vert \underline{A} \right\vert^2$, $m$ and $\omega_o$.
Hint: Use the fact that $k= m \omega_o^2$.

(b)

From the general solution

$$x(t)=x_{\mathrm{eq}}+A \cos\left(\omega_o t+\phi_o\right)$$

compute the average kinetic energy and the average potential energy of the undamped oscillator in terms of $A^2$, and show that they are equal.
Hint: The average values of both $\sin^2$ and $\cos^2$ are $1/2$.

(c)

Using the fact that the work done per unit time (power) against any force is $P = \vec F \cdot \vec v$, show that for the drag force described in lecture, the total energy $E$ (kinetic plus potential) of a damped oscillator obeys

$$\frac{dE}{dt} = -b m v^2.$$ (2)

How does the average of this energy loss $dE/dt$ compare to the average kinetic energy?

(d)

If the damping $b$ is relatively small, the result in (b) that the average kinetic energy is the same as the average potential energy is a very good approximation. Under this approximation, use your result in (c) to show that on average

$$\frac{dE}{dt} \approx -b E.$$ (3)

(e)

The solution to the equation in (d) is that the energy decays exponentially as

$$E(t)=E_o e^{-b t}.$$

Explain (in a brief sentence or two) why the exponent here is ``$-b t$'' whereas the exponent in the general solution for damped harmonic motion is ``$-b t/2$''.