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11.1: Simple Harmonic Motion

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  • 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

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


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

    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 \( \alpha\).