Simple Harmonic Oscillator

solve the questions

2. Energy of a simple harmonic oscillator
The energy topic will be covered later in the course. However, you already have the tools to define 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 defined as Ek = 1 mv 2 . How should it be defined for the pendulum
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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 defined 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 first 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 define 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 definitions above are correct the total energy should be time independent. Show that it is indeed
so (again, there is no friction).

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(g) What is the maximal potential energy of the mass-spring system? What is the maximal kinetic energy?
(h) Use the same definitions 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?
3. Crossing the origin
Show that an over damped harmonic oscillator may cross the origin at most once.