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Erich Poppitz

PHY254 Lecture 5 Kinematics of Oscillations ▯ Readings: Supplementary notes handout. The wikipedia entry for com- plex numbers isn’t bad either. ▯ Kinematics of oscillatory motion and sinusoidal motion. Oscillations - The Basics ▯ An oscillation is a motion that is periodic in time. Imagine some vari- able f(t) describing the motion. It is an oscillation if there is some period T that satis▯es f(t) = f(t + T) = f(t + 2T) = :::: for all t ▯ Oscillations are ubiquitous. They can be very simple (like a sine func- tion), or much more complicated (draw square wave function and more complicated oscillations) ▯ We will begin our study of oscillations by working with sine and cosine functions. Although these are "simple" oscillations, it turns out that they are the basis for the study of all oscillations because of Fourier’s theorem. ▯ Fourier’s theorem: Any period function (as long as its piecewise con- tinuous) can be represented as a sum of sinusoidal functions with ap- propriate amplitudes and frequencies. I won’t prove this now, but we will study this later in the semester. For now, you should keep in mind that any oscillation can be considered a linear combination of sines and cosines. Using sines and cosines to study oscillations ▯ We know that sine and cosine functions are periodic with period 2▯: sin▯ = sin(▯ + 2▯) = sin(▯ + 4▯) = ::: (1) 1 ▯ To relate sin▯ to a time-periodic function, de▯ne the phase: ▯ = !t + ▯; (2) where ! and ▯ are constants. ▯ If the phase advances 2▯, !t + ▯ should advance by T. This means that T = 2▯=!. For a signal Asin(!t + ▯), we de▯ne angular frequency: ! = 2▯=T frequency: ▯ = !=2▯ = 1=T ▯: phase constant. A: amplitude. ▯ Run demo. Describing Oscillations Using Complex Notation ▯ We can connect sinusoidal motion to motion in a circle using the idea of a projection. Show \" ▯ For example, think of an object in uniform circular motion where r = R is a constant and ! is a constant so ▯ = !t + ▯0. At any point in time I can describe the x and y components of the motion as: x(t) = Rcos(!t + ▯ )0 y(t) = Rsin(!t + ▯ 0 2 ▯ So if I only look at a single axis (either the x or the y axis), then the motion is sinusoidal on that axis. This is the same idea as the projection discussed above. ▯ We are going to take advantage of the fact that circular motion pro- jected onto an axis gives sinusoidal motion in order to do calculations of sinusoidal motion. We are going to develop a formalism used in study- ing oscillations and waves that involves describing the motion using \complex numbers". ▯ The idea is as follows: we are going to pretend that 1D sinusoidal mo- tion is actually circular motion but we are going to distinguish between the axis where the "real" motion is occurring and the "imaginary" axis where the imaginary motion (that would result in circular motion if you looked at it in 2D) is occurring. ▯ We’ll take the x axis as the "real" axis and the y axis as the "imaginary axis". Any point along the circular motion could be written in terms of the component on the x axis and the component along the y axis. ▯ Rather than dealing with unit vectors to distinguish the axes we are go- ing to develop a new formalism (i.e. de▯nition of mathematical rules) to study this motion. This formalism will result in making calcula- tions like taking powers, derivatives and integrals really simple. The formalism involves de▯ning a \complex number" z: p z = x + iy; where x and y are real numbers andi ▯ ▯1 (3) ▯ Don’t confuse i with the unit vector in the x direction. It is a new constant which we have de▯ned above (hence the ▯ sign). If I write a unit vector it will always be in bold or with a vector sign or with a hat. ▯ i ▯ p ▯1 should seem like a very strange constant and go against everything you’ve learned about real numbers in mathematics. Thats okay, its suppposed to. We use it in a complex number to distinguish between what is happening on the real axis and what is happening on the imaginary axis. Probably its most important property is that i = ▯1. ▯ Re(z) = x is the real part of z, Im(z) = y is the imaginary part of z. 3 ▯ Complex numbers follow the usual set of math rules. The only trick is that you have to remember the de▯nition of i and hence that i = ▯1. 2 A list of properties of complex numbers is given in the supplementary notes, but here are the basic ones. ▯ If z = x + iy and z = x + iy , then: 1 1 1 2 2 2 { Addition: z1+ z =2(x + 1 ) + 2(y +
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