A) Consider a driven oscillator with dissipation in its steady state, similar to that discussed on page 314 of the textbook. But assume the motor is connected directly to the mass, as sketched above. The maximum force exerted by the motor is F0, which replaces ksD in the textbook. Using the expression for x(t) in the textbook, with D replaced by F0/ks, calculate the sum of the kinetic plus spring potential energy, K+U=(1/2)mv2+(1/2)ksx2. Explain why K+U is not generally constant, unlike the case of an ideal non-driven oscillator with no dissipation. Where does the energy come from and go to? b) At what times during one period is this energy a maximum? Express your answer in tems of ωD and ϕ. What are the values of x and v at those times? HINT: Your answer will depend on whether the driving frequency is greater or less than the resonant frequency (labelled by subscripts D and F in the textbook).

c) Now consider an oscillator with dissipation but no driving force, so that the total force is -ksx(t)-cv(t) where v is the velocity. Derive a formula for the rate of change of (K+U), d(K+U)/dt as a function of t. Express your answer in terms of v(t) and c.