4. This question considers an underdamped 1D oscillator of mass m and spring constant k that is subjected to a damping force of F_drag = -bx. You may assume that the mass is on a horizontal frictionless table, so you only need to consider the spring force and the drag force. At time t = 0, the system is released from rest at x = A.
(a) (4 points) Determine the position x(t) and velocity v(t) of the oscillator as a function of time t.
(b) (2 points) Use part (a) to find an expression for the total energy E(t). (Hint: Some trigonometric double angle identities might help simplify the expression a bit.)
(c) (2 points) Find the rate of energy loss, dE/dt, and show that it is proportional to v^2. (Hint: Use the equation of motion rather than your solution from part b).
(d) (2 points) Is the rate of energy loss a constant? Are there any times when the energy loss rate is 0? Explain why this makes sense.