Simple harmonic motion: the graphs and equations students mix up
Give a student a sketch of a pendulum’s displacement against time and ask for the velocity graph underneath. Quite often the same wave comes back, copied line for line, peaks under peaks. It looks tidy. It also describes a pendulum moving fastest at the exact moment it stops to turn round, which is the one place it isn’t moving at all. Simple harmonic motion is full of small traps like that, and nearly all of them come from learning the graphs as pictures instead of reading them as motion.
In this guide
What makes motion simple harmonic
Plenty of things oscillate. A swing, a guitar string, a car bouncing on its suspension after a speed bump. Simple harmonic motion, or SHM, is one particular, idealised kind of oscillation, and the whole topic hangs off one line in the specification: a = −ω²x.
Read it slowly, because every symbol is doing work. x is the displacement, how far the object is from its resting position. a is its acceleration. ω is the angular frequency, a constant for that oscillator, linked to the period by ω = 2π/T, the same relationship your child met in circular motion. The equation says the acceleration is proportional to the displacement: twice as far out, twice the acceleration.
The minus sign is the part students drop, and it matters most. It says the acceleration always points back towards the middle. Pull a mass on a spring down and it accelerates up; push it up and it accelerates down. Without the minus sign the object would accelerate away from the middle and never come back, which would be a very different, and much shorter, experiment.
A mass on a spring follows this closely. So does a simple pendulum, provided the swings stay small.
The three graphs, and why they’re out of step
Most of the marks in this topic live here.
Your child needs three graphs against time: displacement, velocity and acceleration. All three are smooth waves with the same period. What separates them is where each one starts, and each can be worked out from the one above it.
Velocity is the gradient of the displacement graph. At the top of a displacement peak the graph is flat for an instant, so the velocity there is zero: the object has stopped and is about to come back. Where the displacement graph crosses zero, it’s at its steepest, so the velocity is at its largest. That puts the velocity graph a quarter of a cycle out of step with displacement, or π/2 radians in the language of the course.
Acceleration then follows from a = −ω²x. It’s the displacement graph turned upside down and scaled. Largest displacement, largest acceleration, pointing the other way. Zero displacement, zero acceleration.
Worth knowing cold: at the ends of the swing, displacement is largest, velocity is zero and acceleration is largest. In the middle, displacement is zero, velocity is largest and acceleration is zero.
A student who knows those two lines can rebuild all three graphs in an exam from nothing. A student who memorised the shapes as pictures tends to get one graph right and shift the other two by the wrong amount.
Sine or cosine?
OCR’s Data, Formulae and Relationships booklet gives two versions of the displacement equation, x = A cos ωt and x = A sin ωt, where A is the amplitude, the largest displacement. Students often ask which one is right. Both are. The choice depends only on when the clock starts.
If the object is released from the end of its swing at time zero, displacement starts at its largest value, and so does cosine, so it’s x = A cos ωt. If timing starts as the object passes through the middle, displacement starts at zero, and so does sine, so it’s x = A sin ωt. The question always says which situation it is, usually in the first sentence, and that sentence is the one worth underlining.
Then there’s the calculator. Inside cos ωt the angle is in radians, and a calculator left in degrees mode gives a wrong answer with no warning at all. In this topic more than any other, that’s worth checking before the first calculation.
What the booklet doesn’t spell out
The booklet also gives v = ±ω√(A² − x²), which finds the speed at any displacement. Two things follow from it, and neither is printed for your child.
Put x = 0 into it and the speed is as large as it gets: vmax = Aω. Questions ask for the maximum speed all the time, and students who don’t see that short step go hunting for an equation that isn’t there.
Then the ±. At any displacement between the ends, the object passes through twice in each cycle, once going each way. Same speed, opposite directions.
One more fact isn’t an equation at all, but it’s examined often: the period of a simple harmonic oscillator doesn’t depend on its amplitude. A pendulum that takes two seconds per swing takes two seconds whether its arc is small or a little larger, as long as the swings stay small. The specification calls this an isochronous oscillator, and the word turns up in questions about clocks.
Energy, damping and resonance
In an ideal oscillator, energy only changes form. At the ends of the swing it’s all potential energy, because the object is still for an instant. In the middle it’s all kinetic, because the object is at full speed. The total stays the same, so on a graph of energy against displacement the kinetic and potential curves are two mirror-image parabolas that always add up to a flat line.
Real oscillators lose energy, usually to friction or air resistance, and that’s damping. The amplitude shrinks swing by swing, and heavier damping makes it shrink faster.
Resonance is the last idea in the section, and it’s a favourite for longer written answers. Push an oscillator again and again at a driving frequency equal to its own natural frequency and the amplitude grows very large, because each push arrives at just the right moment to add energy. A child on a swing, pushed at the top of each arc, is the everyday example. More damping makes the resonance peak lower and broader. Your child should be able to sketch that graph of amplitude against driving frequency, explain its shape in words, and give a practical example of resonance, useful or harmful.
A kitchen-table check
Two things you can do without any physics at all.
First, ask your child to sketch the three graphs (displacement, velocity, acceleration) stacked one above the other on the same time axis, from memory. Then ask them to point at the end of a swing on each graph and say what’s happening. If “stopped, largest acceleration” comes back without a glance at the notes, the topic is in good shape. If all the peaks line up vertically, you’ve found the problem, and it’s worth fixing before the mocks.
Second, a two-minute experiment. Tie a bunch of keys to a piece of string about a metre long and let your child time ten swings on their phone, once with a small swing and once with a slightly bigger one that’s still small. The two times come out almost the same. That’s the period not depending on the amplitude, shown on a kitchen chair, and students remember it far better than a sentence in a textbook.
Common questions
What is the defining equation of simple harmonic motion?
a = −ω²x. The acceleration is proportional to the displacement from the equilibrium position and always points back towards it, which is what the minus sign means. ω is the angular frequency, linked to the period by ω = 2π/T.
Why is velocity zero at maximum displacement in simple harmonic motion?
Because at the end of each swing the object stops for an instant before it comes back. On the displacement against time graph this is a peak, where the gradient is zero, and velocity is the gradient of displacement. At the same point the acceleration is at its largest.
Does the amplitude affect the period of a pendulum?
For small swings, no. A simple harmonic oscillator has a period that doesn’t depend on its amplitude, which OCR calls an isochronous oscillator. For a real pendulum this holds well as long as the swings stay small.