The big idea: As something oscillates (a mass on a spring, a swing), its energy keeps swapping between two forms.
Kinetic energy (KE) = the energy of moving. Potential energy (PE) = stored energy — in a spring, the energy stored when it is stretched or squashed.
While they swap, the total energy stays the same (no friction).
New words — amplitude & equilibrium: The equilibrium position is the middle of the swing — the resting point it passes through fastest.
The amplitude (A) is the greatest displacement from that middle — the turning point, where it stops for an instant before coming back.
Where each one peaks: KE is biggest at the centre (moving fastest) · PE is biggest at the ends (the amplitude, momentarily still). At every point in between, KE + PE add up to the same total.
At the very ends of the swing the object stops for an instant, so its KE is zero — at that moment all the energy is potential. So the total energy equals the PE stored at the amplitude. For a mass on a spring of stiffness k stretched to amplitude A, that stored energy is ½kA².
- total energy of the oscillation (J) — stays constant
- spring constant / stiffness (N m⁻¹)
- amplitude — the greatest displacement from the centre (m)
Not in the data booklet — so remember it: E_{total} = ½kA² is not given — you have to know it.
Memory aid: it is just the elastic PE of a spring (½kx²) worked out at the biggest stretch, x = A — the point where the object is still and all the energy is stored.
What the booklet does give you: The data booklet (C.1) gives the SHM relations the energy is built from — the defining rule a = −ω²x and the period link T = 1/f = 2π/ω. From these the total energy can also be written E_T = ½mω²A². Tap the formula to see its booklet badge.
- acceleration (m s⁻²)
- angular frequency (rad s⁻¹) — how fast it oscillates
- displacement from the centre (m)
- period — time for one full oscillation (s)
- frequency — oscillations per second (Hz)
Worked example — total energy of a spring oscillator
A 0.50 kg mass on a spring of stiffness k = 200 N m⁻¹ oscillates with an amplitude of 0.10 m. Find the total energy of the oscillation.
Solution
- Total energy = the energy stored at the amplitude:
- Put in the numbers (k = 200, A = 0.10):
- Work it out — keep the unit:
Final answer
total energy Etotal = 1.0 J — and it stays 1.0 J for the whole oscillation.
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How this is tested: SHM energy questions test the swap and the constant total.
- Paper 1A: a quick MCQ — where the KE or PE is greatest, or what the energy graph looks like. - Paper 2: link the energies — e.g. find the maximum speed by setting the maximum KE equal to the total energy.
Classic trap: the maximum speed is at the centre (where KE peaks), not at the ends — at the ends the object is momentarily at rest.
The key move: At the centre the PE is zero, so the maximum KE equals the total energy:
½mv_{max}² = E_{total} = ½kA². Cancel the ½ and solve for vmax.
IB-style question — maximum speed of the oscillator
The same 0.50 kg mass on the k = 200 N m⁻¹ spring (amplitude 0.10 m, total energy 1.0 J) oscillates back and forth. Find its maximum speed.
Solution
- Maximum speed is at the centre, where all the energy is kinetic, so the max KE equals the total energy:
- Put in the numbers (m = 0.50, Etotal = 1.0):
- Rearrange for vmax²:
- Square-root — keep the unit:
Final answer
maximum speed vmax = 2.0 m s⁻¹ — reached at the centre, where the KE is greatest.