Simple harmonic motion
Simple harmonic motion is the special kind of oscillation where the restoring acceleration is proportional to how far the object is from the middle. Pendulums, masses on springs and atoms in a lattice all come close to it.
Part 1 of 3: Learn it
In short
- SHM: a = −ω²x. The acceleration is proportional to displacement and directed back to the centre.
- Maximum speed ωA at the centre; maximum acceleration ω²A at the ends.
- Spring: T = 2π√(m/k). Pendulum: T = 2π√(l/g). Neither depends on amplitude.
Where this is in your specification
Spec points: AQA 7408 3.6.1.2, OCR A H556 5.3.1, Edexcel 9PH0 Topic 13
| Board | Topic: Circular motion, oscillations and thermal physics |
|---|---|
| AQA 7408 | 3.6.1 and 3.6.2 |
| Edexcel 9PH0 | Topics 6, 9 and 13 |
| OCR H556 | 5.1, 5.2 and 5.3 |
| Higher C857 76 | Our dynamic Universe: special relativity, the expanding Universe (closest topics) |
The defining equation
The minus sign shows that the acceleration always points back towards the equilibrium position. ω = 2πf, so the period is T = 2π/ω.
Displacement, velocity and acceleration
- At the centre (x = 0): speed is greatest, ωA, and acceleration is zero.
- At the ends (x = ±A): speed is zero and acceleration is greatest, ω²A.
- Velocity is a quarter cycle (π/2) out of phase with displacement; acceleration is in antiphase with it.
Springs and pendulums
The pendulum formula only holds for small angles (less than about 10°). Energy swaps between kinetic and potential during each cycle; the total stays the same if there is no damping.
Where in an SHM cycle is the speed greatest?
Show the answer
At the equilibrium position (x = 0).
Part 2 of 3: See it worked
Worked examples
Example 1
Find the period of a simple pendulum of length 0.80 m. (g = 9.81 m/s²)
- T = 2π √(0.80 ÷ 9.81)
- = 2π × 0.2856
Answer: 1.79 s.
Example 2
A 0.50 kg mass on a spring of stiffness 20 N/m oscillates with amplitude 4.0 cm. Find its period and maximum speed.
- T = 2π √(0.50 ÷ 20) = 2π × 0.158 = 0.993 s
- ω = √(k ÷ m) = √40 = 6.32 rad/s
- v max = ωA = 6.32 × 0.040 = 0.253 m/s
Answer: 0.99 s and 0.25 m/s.
Common mistakes
- Forgetting the minus sign in a = −ω²x, or explaining it vaguely.
- Saying the period of a pendulum depends on its mass or amplitude.
- Putting the amplitude in cm into a formula that expects metres.
- Saying acceleration is greatest at the centre. It is zero there.
What happens to the period of a spring oscillator if the mass is multiplied by 4?
Show the answer
It doubles, because T depends on √m.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Where in an SHM cycle is the speed greatest?
At the equilibrium position (x = 0).
What happens to the period of a spring oscillator if the mass is multiplied by 4?
It doubles, because T depends on √m.
Write the condition for SHM in words.
The acceleration is proportional to the displacement from equilibrium and always directed towards it.
Jobs that use this
- Structural engineer (se abre en otra pestaña)
- Physicist (se abre en otra pestaña)
- Mechanical engineer (se abre en otra pestaña)
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
Estos apuntes están en inglés porque siguen los programas de examen del Reino Unido.
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