16.6 Uniform Circular Motion and Simple Harmonic Motion

Uniform Circular Motion and Simple Harmonic Motion

Circular motion and simple harmonic motion (SHM) are deeply connected. When an object moves in a circle at constant speed, its shadow projected onto a flat surface traces out the exact same back-and-forth pattern as a mass on a spring. This relationship lets you use the well-understood math of circles to describe oscillations, and it shows up everywhere from pistons in engines to the motion of pendulums.

Uniform Circular Motion and Simple Harmonic Motion

Shadow projection in circular motion

Uniform circular motion (UCM) is when an object travels at constant speed along a circular path. Even though the speed stays the same, the velocity is always changing direction, which means there's a centripetal acceleration pointing toward the center of the circle. A centripetal force is what keeps the object on that curved path.

Here's where the connection to SHM appears: imagine shining a light straight down onto an object moving in a horizontal circle. The shadow that object casts onto a wall oscillates back and forth along a line. That shadow is undergoing simple harmonic motion.

• The amplitude of the shadow's SHM equals the radius of the circular path.
• The period of the shadow's SHM equals the period of the circular motion.
• Think of a Ferris wheel viewed from the side: a rider moves in a circle, but from your perspective they appear to bob up and down sinusoidally.

Position correlation in circular vs harmonic motion

Consider an object moving in a circle of radius $r$ with a constant angular velocity $\omega$. At any time $t$, the object's angular position is:

$\theta = \omega t$

Now project that position onto each axis:

• x-axis projection: $x = r \cos(\omega t)$
• y-axis projection: $y = r \sin(\omega t)$

Both of these are equations for SHM. The x-projection follows a cosine function, and the y-projection follows a sine function. The two projections are 90° ($\pi/2$ radians) out of phase with each other. When the x-projection is at its maximum displacement, the y-projection is passing through zero, and vice versa.

The displacement $x = r \cos(\omega t)$ is the standard form of the SHM equation, where $x$ is measured from the equilibrium position.

Period and velocity from circular parameters

Because SHM mirrors the timing of UCM, you can pull the period, frequency, velocity, and acceleration directly from the circular motion parameters.

Period and frequency:

• Period: $T = \frac{2\pi}{\omega}$, where $\omega$ is the angular velocity in radians per second.
• Frequency: $f = \frac{1}{T} = \frac{\omega}{2\pi}$

Velocity in SHM comes from differentiating the displacement equation:

1. $v = \frac{dx}{dt} = -r\omega \sin(\omega t)$
2. The maximum speed is $v_{max} = r\omega$, and it occurs as the object passes through the equilibrium position ($x = 0$), where all the energy is kinetic.

Acceleration in SHM comes from differentiating velocity:

1. $a = \frac{dv}{dt} = -r\omega^2 \cos(\omega t)$
2. The maximum acceleration is $a_{max} = r\omega^2$, and it occurs at maximum displacement (the endpoints), where the restoring force is greatest.

Harmonic Oscillators and Forces

A harmonic oscillator is any system that, when displaced from equilibrium, experiences a restoring force that pulls it back. The key feature is that this restoring force is proportional to the displacement and acts in the opposite direction.

The classic example is a mass on a spring. The restoring force follows Hooke's Law:

$F = -kx$

Here, $k$ is the spring constant (a measure of the spring's stiffness, in N/m) and $x$ is the displacement from equilibrium. The negative sign tells you the force always opposes the displacement.

Two additional concepts to know at the introductory level:

• Damping occurs when friction or air resistance gradually removes energy from the system, causing the amplitude to decrease over time. A playground swing slowing down on its own is a good example.
• Resonance happens when you drive an oscillating system at its natural frequency. The amplitude builds up to a maximum because energy is being added in sync with the motion. Resonance can be useful (tuning a radio) or destructive (a bridge vibrating dangerously when wind gusts match its natural frequency).