More simple harmonic motion in a potential In one spatial dimension, a particle

Question

More simple harmonic motion in a potential

In one spatial dimension, a particle moves in a potential V (x). Recall that the potential is related to the force by F x = xV (x).

a) Let V = V0cosh(x/L) where V0, L are constants. Taylor expand V about x = 0, and write the equation of motion keeping only the first nontrivial term (that means, a term in V that gives a force). You will find an equation of the form (d2 /dt2)x + 2x = 0 What is , in terms of the parameters in the potential?

b) What is the general solution for x(t)?

2. Mass on a spring:

A mass m is attached to a spring with spring constant k. At t = 0 the spring is at its rest length (not extended or compressed) and has speed v = v0. Let’s also define 2 = k/m.

a) Solve F = ma to find the motion of the mass. Don’t forget to use the initial conditions.

b) Find the position of the mass at t = /(2) and at t = /(2).

c) What is the period of the oscillations?

d) Suppose that at t = /(4) the mass is at position x0. When will the mass be back at the same position with the same velocity?

Explanation / Answer

More simple harmonic motion in a potential In one spatial dimension, a particle

Related Questions

Navigate

• Browse (All)
• Browse M
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.