What Is Simple Harmonic Motion?
A force that always points home, growing with distance, makes a perfect rhythm.
Lesson at a glanceShow the key points
- Periodic motion repeats after a fixed time (the period); oscillation is to-and-fro motion along the same path.
- SHM: motion on a line with acceleration toward a fixed point and proportional to the displacement: a = −ω²x (12.1).
- Equivalently the net force is a linear restoring force F = −kx (12.2), with k = mω² the force constant.
- The centre of oscillation is the equilibrium position; the maximum displacement on either side is the amplitude A.
- Energy conservation makes the turning points equally far on both sides of the centre.
A child on a swing, a guitar string, the air in a flute, atoms in a crystal: all of them move back and forth around a resting point. The simplest and most important of these motions has a name: simple harmonic motion.
Periodic motion and oscillation
Motion that repeats itself after a fixed interval of time is called periodic, and that interval is the time period. When the body moves to and fro over the same path, the periodic motion is called an oscillation. The Earth going round the Sun is periodic but not an oscillation; a swinging pendulum is both.
What makes it “simple harmonic”
Put the fixed point (the centre of oscillation) at the origin and the line along x:
In an inertial frame a = F/m, so the same statement in terms of force is:
The net force is zero at x = 0, so the centre of oscillation is the equilibrium position. A force that always pushes back toward equilibrium is called a restoring force; SHM is what a restoring force proportional to displacement produces.
A particle in SHM feels a net force of 6.0 N when it is 4.0 cm from the centre of oscillation. Find the force constant.
A block on a spring
Put a block of mass m on a smooth table, attached to a wall by a spring of constant k. At O the spring has its natural length. Pull the block to P, a distance A to the right, and let go. Its displacement x always equals the stretch of the spring, and the spring force is −kx: by definition, this is SHM. Follow one cycle:
- From P to O the force points toward O, so the block speeds up. It is fastest at O.
- Past O the spring is compressed and pushes right, so the block slows, stopping at some point Q.
- Then it is pushed back, speeds up to O, slows again and stops at P. And so on.
The block is momentarily at rest at both P and Q, so all the energy is in the spring at those points. With no friction,
The block swings equal distances either side of the centre. That maximum distance is the amplitude A.
A 0.80 kg body moves in SHM under a force F = −(200 N/m)x. It passes the centre at 3.0 m/s. Find the amplitude.
- 1Periodic motion repeats after a time period; oscillation is periodic to-and-fro motion.
- 2SHM: , or equivalently with .
- 3The centre of oscillation is the equilibrium position; the restoring force always points toward it.
- 4With no friction the body swings equally far either side: the amplitude A.
Check yourself
Keep exploring
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