Simple Harmonic Motion

Note: I explore this topic in more mathematical detail in my Ordinary Differential Equations notes on Undamped Motion.

Simple Harmonic Motion is sinusoidal periodic motion that repeats perfectly over and over. It is a simplified model of motion of objects like a box bouncing on a spring or a simple pendulum, ignoring dampening forces such as friction and making some other simplifying assumptions.

Some basic formulas of a particle exhibiting simple harmonic motion follow.

Here, $x_m$ is the amplitude of the displacement the particle, $\omega$ is the angular frequency of the particle’s motion (i.e. $2 \pi f$ where $f$ is the frequency), and $\phi$ is the phase angle of the particle, i.e. the position in the cycle of the particle at $t = 0$.

Displacement: $x = x_m \cos{(\omega t + \phi)}$

Velocity: $v = \frac{dx}{dt} = - x_m \omega \sin{(\omega t + \phi)}$

Acceleration: $a = \frac{dv}{dt} = - x_m \omega^2 \sin{(\omega t + \phi)}$

In the case the particle has mass $m$ and is moving under the influence of a Hooke’s law restoring force given by $F = -kx$, it has an angular frequency of $\omega = \sqrt{\frac{k}{m}}$ and a period of $T = 2 \pi \sqrt{\frac{m}{k}}.$ Here, $k$ is the spring constant and is given in the unit $N/m$ (newtons per meter.)

Torsion Pendulum

A torsion pendulum is an angular simple harmonic oscilator. It consists of a mass suspended from a fixed wire; the mass can be rotated, causing torsion in the wire, which resists and stores potential energy much like a spring. If the mass is rotated to some angular displacement $\theta$ and released, it will oscilate about that position in angular simple harmonic motion. Rotating the mass through an angle $\theta$ in either direction introduces a restoring torque given by

\[\tau = - \kappa \theta.\]

Here $\kappa$ is called the torsion constant, and depends on the length, diameter, and material of the wire. The formula above is basically the angular version of Hooke’s law. Replacing the spring constant $k$ with $\kappa$ and the mass $m$ with $I$, the rotational inertia of the oscillating mass, we get

\[T = 2 \pi \sqrt{\frac{I}{\kappa}}\]

Simple Pendulum

A simple pendulum is a pendulum with the following simplifying assumptions made:

  • The pendulum bob is assumed to be a point mass.

  • The string/rod is massless

  • There is no air resistance

  • There is no friction

  • The angle of the pendulum relative to vertical is small

  • There are only two dimensions of motion

  • The pendulum has perfectly rigid suspension

The total mechanical energy in a simple pendulum can easily be found by considering that $PE = mgh$, and that when the pendulum bob is at its maximum height, it is stationary and therefore $KE = 0$.

Then, the total mechanical energy of a simple pendulum is $E = PE + KE = mgL(1-\cos{\theta_0})$, where $\theta_0$ is the maximum angular displacement of the pendulum.

Damped Simple Harmonic Motion

If we have damping force that is proportional to the velocity of an object exhibiting simple harmonic motion ($F = -bv$), we now have damped simple harmonic motion. Here, $b$ is the damping constant.

The equation of the displacement of the oscillating particle now becomes:

\[x(t) = x_m e^{-bt/2m} \cos{(\omega't + \phi)}\]

where $\omega’ = \sqrt{\frac{k}{m} - \frac{b^2}{4m^2}}$