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Physics 214 Assignment 1
Concepts:
complex numbers
force law for SHM
complex exponentials
damped oscillations
oscillations
driven oscillations and resonance
restoring forces
decay time and resonance width
simple harmonic motion
oscillator equations
Reading:
AG Lecture Notes on Oscillations (from the website); Y&F, Vol. 1, Chapter 13
Assignment:
Due in lecture on Tuesday, January 27.
Please turn in this sheet stapled to the top of your work.
A.
Math WarmUp Problems
1.
(a)
What are the real and imaginary parts of the function z(t) = r e
i(
ω
t +
φ)
?
(b)
In (a) you assumed that
ω
is real.
But suppose that it has an imaginary component as
well, i.e.,
ω
=
ω
0
+ i
ω
1
.
What is the real part of z(t) in this case?
(c)
From your answer to (a), we can represent a real sinusoidal oscillation using the real part
of a complex exponential, x(t) = Re [z].
Evaluate (i) dz/dt; (iii)
dx/dt = d/dt { Re[z] }; and
(ii) Re {dz/dt}.
(d)
Consider a second order differential equation of the form
a
d
2
x/dt
2
+ b dx/dt + cx = 0
where a, b and c are constants.
Show that x(t) = Re { [z(t)] }, where z(t) is as in (a), is a
solution to this equation for a particular value of
ω
.
Determine that
ω
.
What are its real
and imaginary parts?
Hint: From your answer to part (c), we see that the operations of taking the time
derivative and taking the real part of a complex exponential function of time commute.
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 Spring '07
 GIAMBATTISTA,A
 Exponential Function, Force, Simple Harmonic Motion, Cornell University, Complex number, Euler's formula

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