Defining Simple Harmonic Motion (SHM)
Introduced| English | Meaning |
|---|---|
| simple harmonic motion/ˈsɪmpl hɑːˈmɒnɪk ˈməʊʃn/ | Oscillation in which acceleration is proportional to displacement from equilibrium and directed towards equilibrium. |
Pull a swing back and it always returns
- Pull a child's swing to one side and let go — it rushes back, overshoots, and swings the other way.
- The farther you pull it, the harder it is tugged back toward the middle.
- Any motion with this "always pulled back, in proportion" rule is simple harmonic motion 简谐运动 (SHM).
- It is the physics of springs, pendulums, guitar strings and atoms in a solid.
The defining rule
- SHM happens when the restoring force is proportional to the displacement from equilibrium, and points back toward it.
- In symbols: $F = -kx$ — the minus sign means "back toward the middle".
- Push it further out (bigger $x$) and the pull-back force grows in step.
- This single condition produces the smooth back-and-forth of every oscillator.
Restoring force grows with displacement
Stretch the spring and see the restoring force grow in proportion — the heart of SHM.
A spring has $k = 50\ \tfrac{\text{N}}{\text{m}}$. What is the size of the restoring force at a displacement of $0.2\ \text{m}$, in $\text{N}$?
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$|F| = kx = 50 \times 0.2 = 10\ \text{N}$, pointing back toward equilibrium.
What defines simple harmonic motion?
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SHM needs $F = -kx$: a restoring force proportional to displacement, pointing back to equilibrium.
In $F = -kx$, the minus sign means the force points ____ toward equilibrium.
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The force always opposes the displacement — it points back toward the middle.
Equilibrium and overshoot
- The oscillator has an equilibrium point where the net force is zero.
- Displaced, it is pulled back — but it arrives at equilibrium moving, so it overshoots.
- Then the restoring force slows it, stops it, and pulls it back again.
- The result is an endless, smooth oscillation about the middle.
Why does an SHM oscillator not simply stop at the equilibrium point?
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At equilibrium the force is zero but the speed is greatest, so the oscillator overshoots and swings on.
Where SHM shows up
- A mass on a spring is the textbook example: $F = -kx$ exactly.
- A pendulum swings with SHM too, as long as the angle stays small.
- Atoms vibrating in a crystal, and a plucked string, are SHM as well.
- Whenever a system is nudged from a stable balance, SHM tends to appear.
A pendulum is simple harmonic motion only for small swing angles.
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The restoring force is proportional to displacement only at small angles; large swings are not SHM.
Select all systems that undergo simple harmonic motion (at least approximately).
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Springs, small-angle pendulums and strings all have restoring forces $\propto$ displacement. Constant-speed driving does not.
A pendulum is only approximately SHM — the restoring force is proportional to displacement only for small angles. Swing it too far and the motion is no longer simple harmonic. The mass-on-a-spring is the cleaner ideal.
A spring pulls back with $F = -kx$, where $k = 50\ \tfrac{\text{N}}{\text{m}}$. Find the restoring force at a displacement of $0.2\ \text{m}$.
- $F = -kx = -50 \times 0.2 = -10\ \text{N}$.
The $10\ \text{N}$ force points back toward equilibrium, opposite the displacement.
Simple harmonic motion is oscillation where the restoring force is proportional to displacement and points back to equilibrium: $F = -kx$. The object overshoots equilibrium and swings endlessly. Springs are exact SHM; pendulums are SHM only for small angles.
Write from a restoring force of 12 N
- A spring has k = 80 N/m and is stretched 0.15 m.
- Find the size of the restoring force.
A spring with k = 80 N/m is stretched by 0.15 m. What is the magnitude of the restoring force, and which way does it point?
80 times 0.15 is 12.
What is the size of the restoring force?
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k times x is 80 times 0.15.