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JordanGo
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Classical Mechanics: Finding force, equilibrium points, turning points...
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?
F(x)=-dU/dx (not sure though)
E=K+U
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...
Homework Statement
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?
Homework Equations
F(x)=-dU/dx (not sure though)
E=K+U
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...