11.1: Simple Harmonic Motion

I am assuming that this is by no means the first occasion on which the reader has met simple harmonic motion, and hence in this section I merely summarize the familiar formulas without spending time on numerous elementary examples

Simple harmonic motion can be defined as follows: It a point P moves in a circle of radius $a$ at constant angular speed $\omega$ (and hence period $\frac{2\pi}{\omega}$) its projection Q on a diameter moves with simple harmonic motion. This is illustrated in Figure XI.1, in which the velocity and acceleration of P and of Q are shown as coloured arrows. The velocity of P is just $a\omega$ and its acceleration is the centripetal acceleration $a\omega^{2}$. As in Chapter 8 and elsewhere, I use blue arrows for velocity vectors and green for acceleration.

$P_{0}$ is the initial position of P - i.e. the position of P at time $t=0$ - and a is the initial phase angle. At time $t$ later, the phase angle is $\omega t+\alpha$. The projection of P upon a diameter is Q. The displacement of Q from the origin, and its velocity and acceleration, are

$$y=a\sin(\omega t +\alpha) \label{11.1.1}$$

$$v=\dot{y}=a\omega\cos(\omega t +\alpha) \label{11.1.2}$$

$$\ddot{y}=-a\omega^{2}\sin(\omega t +\alpha). \label{11.1.3}$$

Equations $\ref{11.1.2}$ and $\ref{11.1.3}$ can be obtained immediately either by inspection of Figure XI.1 or by differentiation of Equation $\ref{11.1.1}$. Elimination of the time from Equations $\ref{11.1.1}$ and $\ref{11.1.2}$ and from Equations $\ref{11.1.1}$ and $\ref{11.1.3}$ leads to

$$v=\dot{y}=\omega(a^{2}-y^{2})^{\frac{1}{2}} \label{11.1.4}$$

and

$$\ddot{y}=-\omega^{2}y \label{11.1.5}$$

An alternative definition of simple harmonic motion is to define as simple harmonic motion any motion that obeys the differential Equation $\ref{11.1.5}$. We then have the problem of solving this differential Equation. We can make no progress with this unless we remember to write $\ddot{y}$ as $v\frac{dv}{dy}$ (recall that we did this often in Chapter 6.) Equation $\ref{11.1.5}$ then immediately integrates to

$$v^{2}=\omega^{2}(a^{2}-y^{2}) \nonumber$$

A further integration, with $v=\frac{dy}{dt}$, leads to

$$y = a \sin (\omega t + \alpha) \nonumber$$

provided we remember to use the appropriate initial conditions. Differentiation with respect to time then leads to Equation $\ref{11.1.2}$, and all the other Equations follow.

Exercise $\PageIndex{1}$

Important Problem.

Show that $y=a\sin(\omega t + \alpha)$ can be written

$$y = A \sin \omega t + B \cos \omega t \label{11.1.8}$$

where $A=a\cos\alpha$ and $B=a\sin\alpha$. The converse of these are $a=\sqrt{A^{2}+B^{2}}, \cos\alpha=\frac{A}{\sqrt{A^{2}+B^{2}}}, \sin\alpha=\frac{B}{\sqrt{A^{2}+B^{2}}}$. It is important to note that, if $A$ and $B$ are known, in order to calculate a without ambiguity of quadrant it is entirely necessary to calculate both $\cos\alpha$ and $\sin\alpha$. It will not do, for example, to calculate $\alpha$ solely from $\alpha=\tan^{-1}(\frac{y}{x})$ because this will give two possible solutions for a differing by 180^o.

Show also that Equation $\ref{11.1.8}$ can also be written

$$y=Me^{i\omega t}+Ne^{-i\omega t}, \label{11.1.9}$$

where $M=\frac{1}{2}(B-iA)$ and $N=\frac{1}{2}(B+iA)$ show that the right hand side of Equation $\ref{11.1.9}$ is real.

The four large satellites of Jupiter furnish a beautiful demonstration of simple harmonic motion. Earth is almost in the plane of their orbits, so we see the motion of satellites projected on a diameter. They move to and fro in simple harmonic motion, each with different amplitude (radius of the orbit), period (and hence angular speed $\omega$ ) and initial phase angle $\alpha$.