Show an electron is executing simple harmonic motion

[cherry_ying]

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!

 

[Dadface]

Perhaps you could try the following:
1.Write a value for F when the electron is at a distance r(F1 is proportional to 1/r^2)
2.Write a value for F when displaced from r by a small distance dx(F2 is proportional to 1/(r+ or -dx)^2)
3.Subtract to find the resultant force and use the simplifying assumption that dx is negligible compared to r.

Use F=Ma.If a is proportional to dx and in the opposite direction you have proved SHM.

 

[cherry_ying]

Perhaps you could try the following:
1.Write a value for F when the electron is at a distance r(F1 is proportional to 1/r^2)
2.Write a value for F when displaced from r by a small distance dx(F2 is proportional to 1/(r+ or -dx)^2)
3.Subtract to find the resultant force and use the simplifying assumption that dx is negligible compared to r.

Use F=Ma.If a is proportional to dx and in the opposite direction you have proved SHM.

Thanks for your reply. But isn't the restoring force per unit distance is already given. What's the purpose of finding a resultant force?

Also, how can a be proportional to dx if neither cos or sine is involved? Could you clarify your last statement?

Thanks!

 

[Dadface]

1.The original question is unclear.If r is the radius of the atom then the equation you wrote gives the centripetal(Coulomb) force on the electron.The question states that the electron oscillates about the radius as an "equilibrium radius".The implication is that the electron remains in this radius until momentarily radially displaced then released thereby setting it into vibrations(analogous to displacing then releasing a mass on a spring).I am assuming that the restoring force is equal to the difference between the Coulomb forces at the displacement and equilibrium radii.Did you present the question as it was written?

2.For any particle that moves with SHM the acceleration(a) is directly proportional to its displacement(dx) from a fixed point and directed towards that point.The constant of proportionality is w^2

 

[cherry_ying]

1.The original question is unclear.If r is the radius of the atom then the equation you wrote gives the centripetal(Coulomb) force on the electron.The question states that the electron oscillates about the radius as an "equilibrium radius".The implication is that the electron remains in this radius until momentarily radially displaced then released thereby setting it into vibrations(analogous to displacing then releasing a mass on a spring).I am assuming that the restoring force is equal to the difference between the Coulomb forces at the displacement and equilibrium radii.Did you present the question as it was written?

2.For any particle that moves with SHM the acceleration(a) is directly proportional to its displacement(dx) from a fixed point and directed towards that point.The constant of proportionality is w^2

It seems my prof just changed the question. instead of restoring force at a unit distance, it is now a restoring force at a given radius. Does this make it more clear?

2. but I don't know the proportionality constant though, so by subbing in a specific w^2 it would work, but it doesn't mean that it's SHO.

 

[Dadface]

It's still unclear and contradictory because it implies that there is a restoring force when the electron is in a radius described as an "equilibrium radius"

If a is proportional to -dx then the motion is SHM by definition and w^2 is given by:

a=-omega squared *dx

Anyway cherry ying its nearly midnight here and I'm off to bed

 

[Gokul43201]

Ugh! The question is just bad (in more than one way). It talks about a restoring force, but the equation appears to describe a quantity labeled as "k", which is usually the spring constant in a harmonic oscillator. If the question tells you that the restoring force is described by F=kx, where k=some constant blah (independent of x), then it is automatically telling you that motion is harmonic. The only thing you could perhaps do is calculate the natural frequency from the values of k and m.

Please ask the prof why he says the "force" is given by a certain equation, when the equation does not involve a force anywhere. I think there's something fundamentally wrong with the question, but it might just possibly an error in translation between languages (if there is a translation involved).

 

[cherry_ying]

It's still unclear and contradictory because it implies that there is a restoring force when the electron is in a radius described as an "equilibrium radius"

If a is proportional to -dx then the motion is SHM by definition and w^2 is given by:

a=-omega squared *dx

Anyway cherry ying its nearly midnight here and I'm off to bed

A classmate of mine clarified it for me, it turns out that I just had had to prove K(r0)' = ma, which gives the SHO equation.

Thank you for all your help!

 

[cherry_ying]

Ugh! The question is just bad (in more than one way). It talks about a restoring force, but the equation appears to describe a quantity labeled as "k", which is usually the spring constant in a harmonic oscillator. If the question tells you that the restoring force is described by F=kx, where k=some constant blah (independent of x), then it is automatically telling you that motion is harmonic. The only thing you could perhaps do is calculate the natural frequency from the values of k and m.

Please ask the prof why he says the "force" is given by a certain equation, when the equation does not involve a force anywhere. I think there's something fundamentally wrong with the question, but it might just possibly an error in translation between languages (if there is a translation involved).

It seems that there is no translation error, it's just that our prof decided to take a question from a previous textbook, but make it more complicated by changing some parts (in the original question, it was restoring force per unit distance, but he changed it to restoring force at a given radius, and others)

A classmate showed me what our prof was looking for so now everything's good!

Thanks for your help!

 

[Gokul43201]

Oh well. Guess you have enough to move on.

Incidentally, "restoring force per unit displacement" would have been perfectly fine, but "restoring force at a given radius" is just plain wrong.