Prof. Raghuveer Parthasarathy
University of Oregon; Fall 2007
Physics 351 – Vibrations and Waves
Problem Set 6
Due date:
Friday, Nov. 16, 4pm.
Reading:
French Chapter 5.
A note:
This problem set is shorter than it seems.
Problems 26 are very
similar to one another, which should
cement your understanding of coupled oscillators and should also be useful in analyzing the coupled oscillator systems
that you build.
Problem 1 is not related to coupled oscillators and could have been assigned weeks ago.
(1,
6 pts.
)
Phase space.
It’s often informative to
consider the trajectory of a system in “phase
space,” in which particular physical quantities form
the coordinate axes.
One common set of
quantities is position and velocity.
For example
: let’s
imagine some system in which the position is
given by
()
/
x
ta
t
=
, and the velocity is given by
2
/
vt
x
a t
==
−
±
, where
t
is time and
a
is some
constant.
Therefore
2
/
va
x
=−
– i.e.
vx
is a
quadratic function of
x
.
At time
t
=0, both
v
and
x
are zero, so our parabola “starts” at the origin of
our phase space plot.
The plot looks like:
(a,
3 pts.
)
Consider an
undamped
simple harmonic oscillator.
Prove that the phase space plot (with
axes
x
and
x
±
) is an ellipse.
(Remind yourself of the equation characterizing an ellipse.)
(b,
2 pts.
)
Draw the phase space plot for an undamped oscillator with the initial conditions
0
(0
)
t
x
x
=
=
,
)
0
t
v
=
=
.
Indicate the
t
=0 point on the plot.
(b,
1 pt.
)
What would the phase space plot of a damped oscillator look like?
Draw it, qualitatively –
you don’t have to do any math.
Indicate the “t=
∞
” point on the plot.
(2
, 12 pts
)
Two coupled masses.
Consider two objects (
A
and
B
) of
equal mass
m
connected to each other and to rigid walls by identical
springs of spring constant
k
(see Figure).
Neglect damping, and
assume all motion is in a horizontal plane, so gravity is irrelevant.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 Barzda
 Physics, Normal mode, normal mode frequencies

Click to edit the document details