Defining Simple Harmonic Motion
Introduced| English |
|---|
| simple harmonic motion/ˈsɪmpl hɑːˈmɒnɪk ˈməʊʃn/ |
| restoring force/rɪˈstɔːrɪŋ fɔːs/ |
The back-and-forth that nature loves
- A mass on a spring bobs up and down, over and over.
- A guitar string, a swing, a tuning fork -- all repeat the same dance.
- Pull them aside and something always tugs them back.
- One rule governs every smooth, repeating wobble.
The restoring rule
- Simple harmonic motion 简谐运动 happens when the restoring force 回复力 is proportional to displacement:
- The minus sign means the force always points back toward equilibrium.
- A bigger pull the farther you go -- that is the signature of SHM.
A spring with $k = 20\ \text{N/m}$ is stretched $x = 0.15\ \text{m}$. The magnitude of the restoring force (in N)?
$|F| = kx = 20 \times 0.15 = 3\ \text{N}$.
A motion is simple harmonic when the restoring force is...
$F = -kx$ -- proportional to $x$ and pointing back toward equilibrium.
The minus sign in $F = -kx$ means the force always points back toward equilibrium.
The force opposes the displacement, restoring the mass toward $x = 0$.
Acceleration points home
- Dividing by mass, the acceleration obeys:
- Acceleration is proportional to displacement but opposite in direction.
- The constant $\omega^2 = k/m$ sets how quickly it oscillates.
For SHM with $\omega^2 = 25\ \text{rad}^2/\text{s}^2$ at displacement $x = 0.2\ \text{m}$, the acceleration magnitude (in m/s²)?
$|a| = \omega^2 x = 25 \times 0.2 = 5\ \text{m/s}^2$.
The usual suspects
- A mass on a spring: $F = -kx$ directly.
- A pendulum at small angles: gravity gives a near-linear pull back.
- Any system with a linear restoring force wobbles as SHM.
The restoring force of SHM
Simple harmonic motion needs a restoring force proportional to the displacement and pointing back to the centre.
A $0.5\ \text{kg}$ mass on a spring with $k = 8\ \text{N/m}$ is pulled $0.1\ \text{m}$ and released.
- Restoring force at release: $F = -kx = -8 \times 0.1 = -0.8\ \text{N}$.
- Acceleration: $\omega^2 = k/m = 16$, so $a = -\omega^2 x = -16 \times 0.1 = -1.6\ \text{m/s}^2$.
In SHM, where is the acceleration largest?
$a = -\omega^2 x$ is largest where $x$ is largest -- the extremes.
A pendulum swings as simple harmonic motion only for small angles.
Only at small angles is the restoring pull proportional to displacement.
A pendulum is only SHM for small angles (roughly under $15^\circ$); at large angles the pull-back is no longer proportional to displacement. The minus sign is essential -- without it the force would push the mass away and nothing would oscillate. And acceleration is largest at the extremes (biggest $x$) and zero at equilibrium, the opposite of speed.
Simple harmonic motion is the smooth repeat you get when the restoring force is proportional to displacement and points back home: $F = -kx$, so $a = -\omega^2 x$ with $\omega^2 = k/m$. A spring shows it exactly; a small-angle pendulum shows it approximately. Acceleration peaks at the extremes and vanishes at equilibrium.