Ce

pt
and

Ce

pt
. The solution is the oscillating
line between the two envelope curves. The enve
lope curves give the maximum amplitude of the
oscillation at any given point in time. For example
if you are bungee jumping, you are really inter
ested in computing the envelope curve so that you
do not hit the concrete with your head.
The phase shift
γ
just shifts the graph left or
right but within the envelope curves (the envelope
curves do not change if
γ
changes).
Finally note that the angular
pseudofrequency
(we do not call it a frequency since the solution is
not really a periodic function)
ω
1
becomes smaller
when the damping
c
(and hence
p
) becomes larger.
This makes sense. When we change the damping just a little bit, we do not expect the behavior of
the solution to change dramatically. If we keep making
c
larger, then at some point the solution
should start looking like the solution for critical damping or overdamping, where no oscillation
happens. So if
c
2
approaches 4
km
, we want
ω
1
to approach 0.
On the other hand when
c
becomes smaller,
ω
1
approaches
ω
0
(
ω
1
is always smaller than
ω
0
),
and the solution looks more and more like the steady periodic motion of the undamped case. The
envelope curves become flatter and flatter as
c
(and hence
p
) goes to 0.
108
CHAPTER 2. HIGHER ORDER LINEAR ODES
2.4.4
Exercises
Exercise
2.4.2
.
Consider a mass and spring system with a mass
m
=
2
, spring constant
k
=
3
, and
damping constant
c
=
1
. a) Set up and find the general solution of the system. b) Is the system
underdamped, overdamped or critically damped? c) If the system is not critically damped, find a
c
that makes the system critically damped.
Exercise
2.4.3
.
Do
for m
=
3
, k
=
12
, and c
=
12
.
Exercise
2.4.4
.
Using the mks units (meterskilogramsseconds), suppose you have a spring with
spring constant 4
N
/
m
. You want to use it to weigh items. Assume no friction. You place the mass
on the spring and put it in motion. a) You count and find that the frequency is 0.8 Hz (cycles per
second). What is the mass? b) Find a formula for the mass m given the frequency
ω
in Hz.
Exercise
2.4.5
.
Suppose we add possible friction to
. Further, suppose you do not
know the spring constant, but you have two reference weights 1 kg and 2 kg to calibrate your setup.
You put each in motion on your spring and measure the frequency. For the 1 kg weight you measured
1.1 Hz, for the 2 kg weight you measured 0.8 Hz. a) Find
k
(spring constant) and
c
(damping
constant). b) Find a formula for the mass in terms of the frequency in Hz.
Note that there may be
more than one possible mass for a given frequency.
c) For an unknown object you measured 0.2 Hz,
what is the mass of the object? Suppose that you know that the mass of the unknown object is more
than a kilogram.
Exercise
2.4.6
.
Suppose you wish to measure the friction a mass of 0.1 kg experiences as it slides
along a floor (you wish to find
c
). You have a spring with spring constant
k
=
5
N
/
m
. You take the
spring, you attach it to the mass and fix it to a wall. Then you pull on the spring and let the mass go.