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1. The problem statement, all variables and given/known data

The potential energy between two atoms in a molecule is

U(x) = −1/x^6 +1/x^12

Assume that one of the atoms is very heavy and remains at the origin at rest, and the

other (m = 1) is much less massive and moves only in the x-direction.

(a) Find the force F(x).

(b) Find the equilibrium point x0 and check stability. Give a numerical value for x0

(c) If the system has a total energy E = −0.2, find the turning points, and the period

of oscillation.

(d) If the total energy was E = +0.2, describe the motion of the system.

(e) Find the period for small oscillations around equilibrium. Can the energy in part

(c) be considered “small” in this context?

2. Relevant equations

F(x)=-dU/dx (not sure though)

E=K+U

3. The attempt at a solution

a)

dU/dx = -6/x^7 +12/x^13

b) Set dU/dx to zero and I get:

6x^6-12=0

x=6√2 (I said the negative part is ignored since we are only interested in positive x)

Stability: second derivative of U evaluated at equilibrium point.

d2U/dx2=-42/x^8+156/x^14 = 12.699, so stable

c)

E=K+U=1/2mv^2-1/x^6+1/x^12

v^2=(2x^6-2-0.4x^12)/x^12

turning points when v=0,

5x^6-x^12-5=0

Now I'm stuck...

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# Classical Mechanics: Finding force, equilibrium points, turning points

**Physics Forums | Science Articles, Homework Help, Discussion**