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# 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

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

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

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

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 180o.

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 $$\alpha$$.