The energy topic will be covered later in the course. However, you already have the tools to dene the energy of an harmonic oscillator. We start with a pendulum and then generalize to a spring mass system.
(a) The kinetic energy in any system is dened as Ek = 1 mv 2 . How should it be dened for the pendulum when the motion is circular (i.e., how would you express it in terms of the variables of the problem)?
(b) The gravitational potential energy is proportional to the height of the mass. It is dened up to an arbitrary constant: Hv = mgh. We'll use the convention that the potential energy is zero when θ = 0. Write an expression for the potential energy as a function of θ. Simplify your expression by Taylor expanding the trigonometric function to the rst non-zero term.
(c) Show that if there is no friction the total energy E = Ek + Ev is independent of time. Does the total energy depend on the initial conditions?
(d) What is the average of the kinetic energy over one cycle? What is the average of the potential energy over one cycle?
(e) Now let us dene the kinetic and potential energy for a mass-spring system. Write an expression for the kinetic and potential energy. Hint - use the potential energy you found for the pendulum as a guide.
(f) If your denitions above are correct the total energy should be time independent. Show that it is indeed so (again, there is no friction).
(g) What is the maximal potential energy of the mass-spring system? What is the maximal kinetic energy?
(h) Use the same denitions above but this time assume x(t) is describing the motion of a damped harmonic motion. Is the total energy still a constant of time?
Crossing the origin
Show that an over damped harmonic oscillator may cross the origin at most once.