Physics 505 HW IX 1 Autumn 2005
Physics 505 - Autumn 2005
HW IX
Due 11/30/05
Overview: Recall that solving physics problems is not (just) about solving
differential equations. Use physical reasoning tohelp solve the following
exercises and be certain to show your work. It is also important that you practice
completely solving these exercises, checking for errors as you go along.
1) We want to look more carefully at nonlinear behavior in oscillators as discussed
in Lecture 12. We have in mind a pendulum and so write the equation of motion
as
2 2
0
1 0,
where, as noted in Lecture 12,
1 6
for the pendulum (i.e., from expanding
sin
). In the lecture we focused on a first order analysis, i.e., keeping only terms
up to first order in
(note that here
is dimensionless). Here we want to see
how much the result changes if we keep terms up to second order. Assume the
following form for the solution
2
1 3 5
cos cos 3 cos 5 ,
t A t A t A t
and find the form of the frequency
(in terms of 0
,
and
1
A
) to second order in
. For the
value appropriate to the pendulum, how large can the amplitude
1
A
be before the frequency
differs by more than 20% from the linear result
0
.
2) Consider the motion of a pendulum of length
l
and (velocity dependent)
viscous damping described by
. The equation of motion is
2 2
0 0
sin 0, .
g
l
