Maths › Further Mechanics 2 › Simple harmonic motion
Simple harmonic motion
One differential equation, and every oscillation on the paper is an instance of it. Prove the equation holds, read off ω, and all the standard results follow.
Builds on Newton's laws with a variable force and Differentiating trig, exponentials and logs.
IN THIS TOPIC
- Recognise and prove the simple harmonic condition from an equation of motion.
- Use the standard formulae for displacement, speed and period.
- Solve problems mixing time, displacement and speed.
WHAT YOU PROBABLY THINK
In simple harmonic motion the particle moves fastest at the ends of its path, where the force on it is largest.
One equation, one constant
A particle moves with simple harmonic motion when its acceleration is proportional to its displacement from a fixed point and directed back towards it:
Proving that is the standard first part of a question: set up the equation of motion, measure x from the equilibrium position, and show the acceleration comes out as a negative constant times x. The constant is ω², and everything else follows from it.
The solutions are x = a sin ωt, if timing starts at the centre, or x = a cos ωt, if it starts at an end. The period is T = 2π/ω, independent of the amplitude: that independence is the whole reason pendulum clocks work.
WORKED EXAMPLE
Reading off the standard results
A particle moves with simple harmonic motion of amplitude 0.5 m and period 2 s. Find its maximum speed and maximum acceleration.
ω = 2π/T = π rad/s.
Maximum speed = aω = 0.5π = 1.57 m/s, at the centre.
Maximum acceleration = aω² = 0.5π² = 4.93 m/s², at the ends.
The two maxima happen at opposite places, which is what the opening claim reverses.
Speed against displacement
Eliminating t between the displacement and velocity gives a result worth knowing separately, because it answers questions about position without ever finding the time:
Plotted, that is an ellipse: greatest speed at the centre, zero at the ends, and symmetric about both. It also gives the amplitude immediately from any one pair of matching values of x and v, which is how most questions supply the amplitude without saying so.
Time questions need the sine or cosine form instead. Choose whichever puts t = 0 at the moment the question describes, and remember that the particle passes each interior point twice per cycle, so an inverse sine has two relevant solutions.
YOUR TURN
Mixing the two forms
For the same motion, find the speed when the displacement is 0.3 m, and the time taken to first reach that point from the centre.
Show the working
v² = π²(0.25 − 0.09) = π²(0.16), so v = 0.4π = 1.26 m/s.
Starting at the centre, x = 0.5 sin πt, so 0.3 = 0.5 sin πt gives sin πt = 0.6.
πt = 0.6435, so t = 0.205 s.
The particle returns to x = 0.3 later in the cycle, at t = 1 − 0.205 = 0.795 s, which a question asking for 'the times' would expect as well.
THE EXAM BIT
- Measure x from the equilibrium position, not from a fixed end of the motion.
- To prove simple harmonic motion, produce the equation in the form acceleration = −(constant)x and name ω².
- Use v² = ω²(a² − x²) whenever time is not mentioned; it saves several lines.
- Choose sine or cosine to match where the particle is at t = 0, and say which you chose.
CHECK YOURSELF
A particle moves with simple harmonic motion of amplitude 2 m and ω = 3 rad/s. Find its speed at a displacement of 1 m.
Show a hint
One standard formula.
Show the answer
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Simple harmonic motion is the equation acceleration = −ω²x, with solutions a sin ωt or a cos ωt and period 2π/ω whatever the amplitude.
Speed and displacement are linked by v² = ω²(a² − x²): fastest at the centre, at rest at the ends.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Recognise and prove the simple harmonic condition from an equation of motion.
- Use the standard formulae for displacement, speed and period.
- Solve problems mixing time, displacement and speed.
Open the full revision checklist to see every objective in the course in one place.
No animated video for this topic yet; these notes stand alone.