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Lecture

# L06_supplementary.pdf

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University of Toronto St. George

Physics

PHY354H1

Erich Poppitz

Winter

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Lecture 6: Supplementary notes
Sinusoidal functions of time
Now that we have gotten familiar with the basic properties of complex num-
bers, how can we apply this to oscillating particles? Think about a particle
that is moving along the x axis according to x(t) = Acos(!t + ▯), where
A > 0, !, and ▯ are real. We can think of this x(t) as the real part of a
complex function that can be written in exponential form:
x(t) = Acos(!t + ▯)
= ARefcos(!t + ▯) + isin(!t + ▯)g (Imaginary component chosen strategically)
n i(!t+▯)
= ARe e De Moivre thm.
n o
= Re Ae i(!t+▯) Since A is real.
Writing x(t) in this way has a lot of advantages. First, in many cases
it makes it easy to calculate time derivatives and therfore calculate velocity
and acceleration. If x(t) = Refzg, then v = x _ = Refz_g, and a = Refz g.
If z = Ae i(!t+▯, then _ = i!z and z = ▯! z. Then it follows that v =
▯!Asin(!t + ▯) and a = ▯! Acos(!t + ▯). Of course, these results can be
derived using sinusoidal functions directly, but the fact that derivatives are
taken directly on exponentials can simplify the math in complicated problems
considerably.
Second, if we have two sinusoidal signals we wish to add, we can use the
geometric/vector techniques of addition of complex numbers we saw in Lec-
ture 5 Supplementary Notes once the signals are in polar form. To illustrate
this, consider the following example, which involves the superposition of two
signals with the same frequency ! = 3.
▯ Example: Consider two signals x (t)1= 5cos(3t + ▯=6) and x (t) 2
6sin(3t + ▯=4). Find the superposition (the sum) of these two signals.
▯ Answer: First, we need to write both signals in complex form. We
n i(3t+▯=6)
can see quickly that 1 = Re 5e . For 2 , we use :
n o n o n o n o
i▯ i▯ ▯i▯=2 i▯ i(▯▯▯=2)
sin▯ = Im e = Re ▯ie = Re e e = Re e : (1)
n o
This means that x = Re 6e i(3t▯▯=4, and because
2
Refz 1 + Refz 2 = Refz +1z g;2
1 we can write
i(3t+▯=6) i(3t▯▯=4)
x1+ x 2 = Ref5e + 6e g
i3t i▯=6 ▯i▯=4▯
= Refe 5e + 6e g
Within the curled braces, we have the product of two complex numbers.
The one in parentheses is a time independent sum of a vector with
magnitude 5 and orientation ▯=6 and a vector with magnitude 6 and
orientation ▯▯=4. The vector sum is shown in the ▯gure below.
2.5
z1
2.0 z2
z1+z2
1.5
1.0
0.5
imaginary
0.5
1.0
1.5
2.0 1 2 3 4 5 6 7 8 9
real
The factor e i3trepresents a rotation of this vector (the dashed line in
the ▯gure above) by an angle 3t at time t. Thus, the resultant rotates
with angular frequency 3.
To ▯nish this example, we have
( p p p !)
i3t 5 3 + 6 2 5 ▯ 6 2
x1+ x 2 = Re e + i
n

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