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cherry_ying

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## Homework Statement

When the electron in a hydrogen atom bound to the nucleus moves a small distance from its equilibrium position, a restoring force at a given radius is given by:

k = e^2/4*pi*epsilon*r^2

where r = 0.05 nm may be taken as the radius of the atom (the equilibrium radius of the electron relative to the proton). Show that the electron can oscillate about this radius, executing simple harmonic motion and find the natural frequency w0.

e = 1.6 *10^-19C; mass of electron = 9.1 * 10^-31 kg; epsilon = 8.85 * 10^-12 N^-1 m^-2 C^2 and c = 3*10^8 m/s.

## Homework Equations

restoring force of a spring = -kx

simple harmonic oscillation: ma+kx=0

natural frequency = sqrt(k/m), k = spring constant

## The Attempt at a Solution

[STRIKE]I am assuming that the -kx = restoring force = k = e^2/4*pi*epsilon*r^2

Taking second derivative of restoring force, k, we would get kx'' = ka = 6*e^2/4*pi*epsilon*r^4

(assuming that all the constants combine together gives the spring constant and the radius is the x in the restoring force of a spring equation, spring constant = e^2/4*pi*epsilon)

so ma+kx =0 => m*6*r^-4 + e^2/4*pi*epsilon*r^4, which doesn't equal to zero.

Also, shouldn't Simple Harmonic Oscillation involve cos? But I can't seem to see where cos would fit in here.[/STRIKE]

After reading the question again, I realized that the restoring force per unit distance k is the spring constant. And if we use ma+kx=0, and assuming x is in the form of cos(theta) then it would work. However, if I do just plug it in, at the end, it would equal to zero if the natural frequency equal to sqrt(k/m). But I assume that the question is asking me to prove it's Simple Harmonic Oscillation without using the natural frequency. So, how do people know the simple harmonic motion is is in the form of cos or sine?

(Also, our professor told us to use linear approximation, but I have no idea how that would work)

Any help would be appreciated, thanks!

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