Defining Simple Harmonic Motion
Introduced| English | Meaning |
|---|---|
| simple harmonic motion/ˈsɪmpl hɑːˈmɒnɪk ˈməʊʃn/ | Oscillation in which acceleration is proportional to displacement from equilibrium and directed towards equilibrium. |
| restoring force/rɪˈstɔːrɪŋ fɔːs/ | A force acting towards an object's equilibrium position after displacement. |
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.
In SHM the acceleration always points back towards equilibrium, opposite the displacement
A spring with $k = 20\ \text{N/m}$ is stretched $x = 0.15\ \text{m}$. The magnitude of the restoring force (in N)?
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$|F| = kx = 20 \times 0.15 = 3\ \text{N}$.
A motion is simple harmonic when the restoring force is...
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$F = -kx$ -- proportional to $x$ and pointing back toward equilibrium.
The minus sign in $F = -kx$ means the force always points back toward equilibrium.
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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)?
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$|a| = \omega^2 x = 25 \times 0.2 = 5\ \text{m/s}^2$.
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?
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$a = -\omega^2 x$ is largest where $x$ is largest -- the extremes.
A pendulum swings as simple harmonic motion only for small angles.
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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.
Write from a restoring force of 5 N
- A spring has k = 50 N/m and is stretched 0.10 m.
- Find the size of the restoring force.
A spring with k = 50 N/m is stretched by 0.10 m. What is the magnitude of the restoring force, and which way does it point?
50 times 0.10 is 5.
What is the size of the restoring force?
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k times x is 50 times 0.10.