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University of Toronto St. George
Erich Poppitz

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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