A particle of unit mass moves along the positive x axis under a force: 36 f = x3 9 x2 a. Find the potential as a function of x; what is the equilibrium point? Sketch the force and the potential and show that the motion is (i) oscillatory if E < 0 and (ii) unbounded for large x if E > 0.
A particle of unit mass moves along the positive x axis under a force: 36 f = x3 9 x2 a. Find the potential as a function of x; what is the equilibrium point? Sketch the force and the potential and show that the motion is (i) oscillatory if E < 0 and (ii) unbounded for large x if E > 0.
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Transcribed Image Text:HI1. A particle of unit mass moves along the positive x axis under a force:
36
f =
x3
x2
a. Find the potential as a function of x; what is the equilibrium point? Sketch the force and the potential
and show that the motion is (i) oscillatory if E < 0 and (ii) unbounded for large x if E > 0.
b. Given that E < 0 find the turning point (s) as a function of E.
c. Find the period of the oscillatory motion.
hint: You will need the following integral:
dx
V(x+ – x)(x – x_)
it
?(0)-
to compute it, use the following substitution: x = x_ cos?(0) + x+ sin² (0) .
d. Now expand the potential around the equilibrium, take the harmonic approximation (keeping up to
quadratic terms near the minimum) and find the period in this approximation.
e. Show that your results in c and d are consistent for small oscillations.
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