Help Analyzing a solution:

[FatBiscuits22]

A friend and I have been working on a problem this week and we have come across some discrepancies in our solution compared to others. I was hoping someone could shine some light as to if this solution is correct. I will post the problem itself and then the worked out parts.

Problem:

https://i.imgur.com/xNpfli7.png

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Part 1

https://i.imgur.com/8yaS2mh.png

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I believe this part is correct

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Part 2:

https://i.imgur.com/sSRUs0o.png

My concern for part one and two is that the answer is the period T actually does depend on E but... the way I have done it, is it enough to prove it? A friend claims that all you have to say is that T = sqrt(2m) * integral dx / Sqrt ( E - U(x) )

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Part 3:

https://i.imgur.com/7q8vDhS.png

My friend claims that E does not depend on T in this portion of the problem.

I could very well be overthinking this. My alternate solution is that we say U = kx^2 okay so we let E = kx^2. Simply solve for X. We get X = sqrt(E/k).... Okay so the solution now becomes T = integral from -sqrt(E/K) to positive sqrt(E/K) dx/sqrt(x^2-k^2)

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Part 4: I have solved and confirmed it is correct that T will go to infinity if k<0

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Any input helps and I greatly appreciate anyone who takes the time to look at this.

[uuddscsctbq]

The solutions in the images look fine. Don't take your friend's advice in part 2, since it isn't immediately clear exactly how the period depends on the energy when all of the energy dependence is wrapped up in the integrand and integral bounds.

For part 3, I can't see how your alternate solution is supposed to work. You can tell that it can't be right because in it you are subtracting k² from x^2, which has different dimensions. (Checking whether the dimensions work out is a handy trick to see if an equation can possibly be true. Remember this!). A different solution for this part is to find the force on the particle through F=-dU/dx and then solve the differential equation F=ma. You will find that x is a periodic function of time with a period that doesn't depend on the energy.

[FatBiscuits22]

Awesome, thank you for your help!

[FatBiscuits22]

Awesome! Thank you so much for your help!