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.

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