What is the Angular Frequency of Small Oscillations for a One-Dimensional Mass?

[Borus Ken]

Homework Statement

This is the problem verbatim:

The Potential energy of a one-dimensional mass m at distance r from the origin is

U(r) = U₀ ((r/R) +(lambda^2 (R/r))

for 0 < r < infinity, with U₀ , R, and lambda all positive constants. Find the equilibrium position r₀. Let x be the distance from equilibrium and show that, for small x, the PE has the form U = const + 1/2 kx^2. What is the angular frequency of small oscillations?

Homework Equations

The Attempt at a Solution

[/B]
I have solved for the equilibrium position by taking the first derivative and setting that equal to zero to find that position to be lambda R.

My problem now is that I cannot figure out how to arrange the equation in the aforementioned form. I have taken the Taylor Polynomial of U(r) and eliminated the first few terms leaving the second derivative multiplied by x^2/2 which is obviously where that portion in the above equation comes from. However, I do not get a constant if I sub r0 in ignoring x because of it being small. I really have tried many different attempts and cannot figure it out.

 

[TSny]

Welcome to PF!

Please show your attempt at the Taylor expansion so that we can see if there are any errors in your work. I don't understand the following statement:

I do not get a constant if I sub r0 in ignoring x because of it being small.

 

[Borus Ken]

Thanks for the reply and I apologize. My attempt is as follows.

U( r) = U ( r) + U'(r)x + (1/2)U''(r)x^2... Ignoring the following terms because x is already small and x^n where n >2 is negligible. Also, from what I have gathered the first term can be ignored and U'(r)x near equilibrium will be close to zero. Therefore the only term left is (1/2) U''(r)x^2. For the second derivative of the equation I get:

U''(r) = U₀ 2(lambda^2) R/r^3.

Subbing in r₀ + x gives a nasty equation. However, If I ignore x because it is small ( not sure if I can do that, I was just trying to find a solution) I only get a term proportional to x^2 or akin to 1/2 kx^2 rather than that plus a constant. So I am stumped to say the least.

 

[TSny]

OK, you have the right idea, but you need to be careful with exactly how you write the expansion. You wrote

U( r) = U ( r) + U'(r)x + (1/2)U''(r)x^2...

But note how this doesn't make sense as written. On the left you have $U(r)$. But the first term on the right side is also $U(r)$. Clearly, something's wrong here.

Review how to do a Taylor expansion about a point. For example, see this link https://en.wikipedia.org/wiki/Taylor_series#Definition

You are dealing with a function of $r$, so you should replace all the $x$'s in the link with $r$'s. You also need to think about the choice of the point $a$ in the link. That is, decide what value of $r$ that you want to "expand about".

 

[Borus Ken]

Thank you TSny.

I haven't reviewed Taylor Series for about a year and after reviewing them I was quickly able to solve the problem. I appreciate your help.