# Simple Harmonic Motion

The equation that fully describes simple harmonic motion is:

<dmath> x(t) = A\cos (\omega t + \varphi ) \ , </dmath>

where:

• ${\displaystyle A}$ is the amplitude;
• ${\displaystyle \omega }$ is the frequency, and ${\displaystyle \omega ={\frac {2\pi }{T}}}$, where ${\displaystyle T}$ is the period;
• ${\displaystyle t}$ is the time variable;
• ${\displaystyle \varphi }$ is the phase.

The choice of using cosine instead of sine is purely arbitrary, the important concept is that the motion is sinusoidal. In fact, the two functions are interchangeable, they only differ by a translation factor of ${\displaystyle {\frac {\pi }{2}}}$, as the following relationship shows: ${\displaystyle x(t)=A\cos(\omega t-{\frac {\pi }{2}})=A\sin(\omega t)\ .}$

Let's now find the equations for velocity and acceleration, respectively by deriving once and twice the expression for ${\displaystyle x(t)}$:

<dmath> v(t) = \frac{ d x(t)}{ d t} = -\omega A\sin (\omega t + \varphi ) \ , </dmath>

<dmath> a(t) = \frac{ d^2 x(t)}{ d t^2} = -\omega ^2 A\cos (\omega t + \varphi ) \ .</dmath>

Here is a graph where displacement, velocity and acceleration are plotted on the same axis: note how each one is just translated by ${\displaystyle {\frac {\pi }{2}}}$ on the right from its derivative (the phase is set to zero for convenience).

It is very important to note that the expression for the acceleration can be rewritten as:

<dmath> \frac{ d^2 x(t)}{ d t^2} = -\omega ^2 x(t) \ .</dmath>

This is a common and easily recognisable trademark of SHM; when solving problems, finding a relation such as ${\displaystyle a(t)=-kx(t)}$ where ${\displaystyle k}$ is some constant (it can also be ${\displaystyle F(t)=-k'x(t)}$, because of Newton's second Law ${\displaystyle F=ma}$) undoubtedly tells you that the motion is simple harmonic.

You can also find the frequency straight away: ${\displaystyle \omega ={\sqrt {k}}}$ (or ${\displaystyle \omega ={\sqrt {\frac {k'}{m}}}}$ in the second case).

This relationship shows that a ”'necessary and sufficient”' condition for having simple harmonic motion is that the acceleration must be:

• proportional to the displacement,
• always directed opposite to the motion.

Here are some applications of simple harmonic motion:

## Mass on a spring

A mass on an ideal spring moves with simple harmonic motion. Let's start from Hook's law: ${\displaystyle {\vec {F}}=-k{\vec {x}}}$ (${\displaystyle k}$ is the spring constant).
For Newton's second Law:

${\displaystyle m{\vec {a}}=-k{\vec {x}}\ ,}$

which can be rearranged in the familiar SHM formula:

<dmath> \vec{a} = - \frac{k}{m} \vec{x} \ , </dmath>

where the frequency is:

<dmath> \omega = \sqrt{\frac{k}{m}} \ .</dmath>

## Pendulum

The motion of a pendulum can be approximated to a simple harmonic motion, basing on the assumption that the angle of swing ${\displaystyle \theta }$ is small, so that ${\displaystyle \sin \theta \approx \theta }$.
Here is a diagram that shows the forces involved and the resultant acceleration, in a conveniently chosen coordinate system:

The tension cancels out the y-component of the weight, so the net force is:
${\displaystyle F=-mg\sin \theta }$

${\displaystyle \sin \theta =\theta }$ (for small angles ${\displaystyle \theta }$);
${\displaystyle a=\alpha L}$ where ${\displaystyle \alpha }$ is the angular frequency and ${\displaystyle L}$ is the length of the pendulum;

${\displaystyle \Rightarrow m\alpha L=-mg\theta }$
and we finally arrive to the characteristic equation of SHM:

<dmath> \frac{d^2 \theta }{dt^2} = -\frac{g}{L} \theta \ ,</dmath>

from which we can work out the frequency (note: it does not depend on the mass!):

<dmath> \omega = \sqrt\frac {g}{L} \ .</dmath>