- #1
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
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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.