IB Physics HLTopic 3 — Oscillations & WavesPaper 1 & 2KE ↔ PE interchange~11 min read
Energy Changes in Simple Harmonic Motion
Watch a pendulum swing and you’re watching energy shuffle back and forth. At the top of each swing it pauses, holding all its energy as stored potential. Rushing through the bottom it’s moving fastest, all that energy now kinetic. The clever part: if nothing steals energy away, the total never changes — it just keeps trading between the two forms, over and over.
📘 What you need to know
SHM continually swaps energy between kinetic energy (KE) and potential energy (PE)
At the extremes (amplitude): KE = 0 and PE is maximum — the object is momentarily still
At the equilibrium position: KE is maximum and PE = 0 — the object is moving fastest
The total energy stays constant (if there’s no damping): Etotal = KE + PE
For a spring the PE is elastic; for a pendulum it’s gravitational — but the pattern is the same
On graphs, total energy is a flat horizontal line; KE and PE curves always add up to it
The energy interchange
Every oscillation is a continuous trade between two energy stores. Which store is “full” depends entirely on where the object is in its swing. Let’s follow a mass on a spring through one journey:
🔄 One swing, following the energy
At the extreme (amplitude): the object is momentarily stationary, so KE = 0. All the energy is stored as PE. This is the turning point.
Moving back towards the centre: the object speeds up, so KE rises while PE falls.
At equilibrium (centre): the object is moving fastest, so KE is at its maximum and PE = 0.
Past the centre: it slows down again, KE falls and PE builds back up — until it stops at the other extreme, and the cycle repeats.
The type of PE depends on the system. For a mass on a spring it’s elastic potential energy stored in the stretched or squashed spring. For a pendulum it’s gravitational potential energy, because the bob rises higher at the ends of its swing. Different stores, but the exact same back-and-forth pattern.
Total energy is conserved
Here’s the key principle. If no energy is lost to friction or air resistance (we call this undamped motion), then the total energy stays constant. Energy is never created or destroyed — it’s just transferred from one store to the other and back. This is the law of conservation of energy in action.
Total energy of an SHM systemEtotal = KE + PE = constant
Because the total is fixed, the two stores are perfectly linked: whenever KE goes up, PE goes down by exactly the same amount, and vice versa. Knowing one plus the total instantly gives you the other.
WE 1
An oscillator has a total energy of 80 mJ. At one point in its swing its kinetic energy is 30 mJ. What is its potential energy at that instant?
Step 1 — total energy is conserved: E = KE + PE
so PE = E − KE
Step 2 — substitutePE = 80 − 30 = 50 mJPE = 50 mJThe two always add to the total. No need for masses or speeds — conservation does it all.
Energy against displacement
The clearest picture comes from plotting energy against displacement. As the object moves from the centre out to the amplitude and back, KE and PE trace out two mirror-image curves under a flat total-energy line.
PE (blue) is a U — zero at the centre, biggest at the extremes. KE (red) is its mirror — biggest at the centre, zero at the extremes. Together they always sum to the flat total.
Notice how at any displacement, the height of the red KE curve plus the height of the blue PE curve reaches exactly the total-energy line. The two curves cross at the halfway height — the point where kinetic and potential energy are momentarily equal.
WE 2
An oscillator has a total energy of 0.48 J. When it is halfway to the amplitude (displacement = x₀/2), what are its kinetic and potential energies? (Use the fact that PE is proportional to x².)
Step 1 — PE grows with x², and PE = total at x = x₀
so PE / E = (x / x₀)² = (1/2)² = 1/4
Step 2 — find PEPE = ¼ × 0.48 = 0.12 JStep 3 — KE is the restKE = 0.48 − 0.12 = 0.36 JKE = 0.36 J, PE = 0.12 JHalfway out in distance is only a quarter of the way in energy — because energy depends on x squared.
Energy against time
We can also plot the energies against time as the oscillation runs. Both KE and PE rise and fall smoothly, always in opposite directions, while the total stays pinned as a flat line on top.
KE (red) and PE (blue) rise and fall in opposite step. When one peaks the other is zero. Their sum — the total — is always the flat orange line. The energy cycles twice per oscillation.
Spot the neat detail: the energy graphs complete two full cycles for every one oscillation. Why? Because the object reaches maximum displacement twice per swing — once on each side — and PE peaks at both. Also, energy is always positive: you’ll never see these curves dip below the axis.
Key positions summarised
Position
Speed
Kinetic energy
Potential energy
Equilibrium (centre, x = 0)
Maximum
Maximum
Zero
Halfway out (x = x₀/2)
Moderate
¾ of total
¼ of total
Amplitude (extreme, x = x₀)
Zero
Zero
Maximum
At the extreme PE max, KE = 0
speeds up PE → KE
At equilibrium KE max, PE = 0
slows down KE → PE
Other extreme PE max, KE = 0
💡 Top tips
KE and PE are always opposite: where one is maximum, the other is zero.
KE max at the centre, PE max at the extremes — link it to speed (fastest in the middle, still at the ends).
Total energy is a flat horizontal line on any SHM energy graph — use a ruler to draw it.
On an energy–time graph the energies cycle twice per oscillation; on energy–displacement the curves are a U (PE) and an n (KE).
Energy is never negative — a curve dipping below the axis is wrong.
⚠ Common mistakes
Putting KE maximum at the extremes — it’s maximum at the centre, where the object moves fastest
Drawing the total-energy line as a curve — if undamped, it’s a straight horizontal line
Forgetting the energy graphs cycle twice per oscillation, not once
Drawing negative energy values — KE and PE are always positive
Assuming PE is halfway when displacement is halfway — PE depends on x², so it’s only a quarter
Quick recap: SHM constantly swaps kinetic and potential energy. KE is maximum at equilibrium (PE = 0) and zero at the extremes (PE maximum), and the two always add to a constant total if the motion is undamped. On an energy–displacement graph, PE is a U and KE is an n under a flat total line; on an energy–time graph, both cycle twice per oscillation and never go negative.
You now understand how the energy moves in SHM — the qualitative picture. The next step is to put real numbers on it: the equations that let you calculate kinetic, potential and total energy from the mass, angular frequency, amplitude and displacement. That’s the next page: calculating energy changes in SHM.
Want the energy picture to click?
Book a free meeting and we’ll work through the KE–PE interchange, the energy graphs and past-paper questions together.
