Representing and Analyzing SHM
SHM draws a perfect sine wave in time
- Track an oscillating mass and plot its position against time — you get a smooth, repeating wave.
- Not a jagged zig-zag, but the graceful curve of a sine or cosine.
- That shape is the fingerprint of simple harmonic motion, and it lets us predict every instant.
- Reading it tells you position, velocity and acceleration at a glance.
The equation of motion
- Starting from the maximum, position follows $x(t) = A\cos(\omega t)$.
- $A$ is the amplitude (the maximum displacement) and $\omega = 2\pi f$ is the angular frequency.
- After one period $T$ the curve repeats exactly.
- Feed in a time and the formula gives the exact position.
Position around a cycle
Step through the phase of the oscillation and see how position and energy vary across a cycle.
An oscillator is $x(t) = 0.1\cos(4t)$ (metres). What is its position at $t = 0$, in metres?
$x(0) = 0.1\cos 0 = 0.1\ \text{m}$ — the maximum displacement (amplitude).
In $x(t) = A\cos(\omega t)$, the symbol $A$ is the ____.
$A$ is the amplitude — the maximum displacement from equilibrium.
Where it's fastest and where it accelerates
- Speed is greatest at the middle (equilibrium) and zero at the ends (turning points).
- Acceleration is greatest at the ends and zero at the middle.
- This is because acceleration follows the force: $a = -\dfrac{k}{m}x$, biggest where $x$ is biggest.
- So velocity and acceleration are "out of step" — one peaks where the other is zero.
Where is an SHM oscillator moving fastest?
Speed is greatest at the middle, where the acceleration (and force) is zero.
Where is the acceleration of an SHM oscillator greatest?
$a = -\tfrac{k}{m}x$ is largest where $x$ is largest — at the extremes.
A mass starts at the far right. Order what happens next, first at the top.
It accelerates toward the middle (max speed there), then decelerates to a stop at the other extreme.
Position, velocity, acceleration
- If position is a cosine, velocity is a (negative) sine, and acceleration is a (negative) cosine.
- Each is shifted a quarter-cycle from the last.
- The acceleration is always opposite the displacement — that is the SHM condition again.
- Together the three graphs tell the whole story of the motion.
In SHM the acceleration always points opposite to the displacement.
$a = -\tfrac{k}{m}x$: the minus sign means acceleration opposes displacement.
In SHM, velocity and acceleration do not peak at the same place. Velocity is largest at the centre (where acceleration is zero); acceleration is largest at the extremes (where velocity is zero). Mixing these up is a classic error.
An oscillator has amplitude $A = 0.1\ \text{m}$ and angular frequency $\omega = 4\ \tfrac{\text{rad}}{\text{s}}$, starting at the maximum.
- Position: $x(t) = 0.1\cos(4t)$.
- At $t = 0$: $x = 0.1\cos 0 = 0.1\ \text{m}$ (the maximum), where the speed is zero.
SHM position follows $x(t) = A\cos(\omega t)$ — a sinusoid of amplitude $A$ and angular frequency $\omega = 2\pi f$. Speed peaks at the centre, acceleration peaks at the extremes, and acceleration always points opposite the displacement.