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Simple harmonic motion (SHM) is one of the most important ideas in physics. It describes the way anything wobbles: a mass on a spring, the pendulum of a clock, a guitar string, the atoms in a crystal, even molecules of carbon dioxide absorbing infrared radiation in the atmosphere. Whenever a system is slightly disturbed from equilibrium and then released, its motion — at least for small displacements — is almost always simple harmonic.
This lesson introduces the defining equation of SHM, explains what makes an oscillation "simple harmonic", and sets up the mathematical machinery we will use throughout OCR A-Level Physics A Module 5.3.
An oscillation is any repeating to-and-fro motion about an equilibrium position. The object returns through the same sequence of positions at regular intervals. Familiar examples:
Each full repetition is a cycle. The time for one complete cycle is the period T, and the number of cycles per second is the frequency f = 1/T. Both of these have exactly the same meaning as in circular motion — indeed there is a deep mathematical link between circular motion and SHM which we will exploit in the next lesson.
What makes something oscillate rather than just sit still? It needs a restoring force — a force that always pushes or pulls the object back towards its equilibrium position.
Consider a mass on a spring. If you pull the mass to the right, the spring stretches and pulls it back to the left. If you push the mass to the left, the spring compresses and pushes it back to the right. Either way, the force is always directed toward the equilibrium point.
graph LR
A[Left of equilibrium] -- restoring F to the right --> E[Equilibrium]
E -- restoring F to the left --> B[Right of equilibrium]
If the restoring force is proportional to the displacement, something magical happens: the motion turns out to be simple harmonic, with sinusoidal solutions and a fixed period that does not depend on the amplitude. This is the defining condition of SHM.
An object undergoes simple harmonic motion if, and only if, its acceleration satisfies
a = −ω²x
where x is the displacement from equilibrium and ω is a positive constant called the angular frequency (units: rad s⁻¹). Read the equation in English:
"The acceleration is proportional to the displacement and directed towards the equilibrium position."
The minus sign is essential — it encodes the restoring nature of the force. Without it you would have an unstable system that accelerates away from equilibrium.
For a system to execute SHM, two conditions must be satisfied:
|a| ∝ |x|).x).Both conditions together are encoded in the single equation a = −ω²x.
Exam Tip: OCR very often asks "State the two conditions for a body to execute simple harmonic motion." Memorise the two bullet points above. A common full-mark answer: "(i) The acceleration is proportional to the displacement from a fixed equilibrium position; (ii) the acceleration is always directed towards this equilibrium position."
There are plenty of periodic motions that are not simple harmonic. A ball bouncing elastically between two walls has a restoring force only at the walls — zero everywhere else. A pendulum swung through a huge amplitude has a restoring force proportional to sin θ, not to θ, and so is only approximately SHM.
"Simple" in SHM refers to the linear relationship a ∝ x. Any more complicated relationship is called anharmonic, and its period generally depends on amplitude.
The constant ω in the defining equation plays exactly the same role as the angular velocity in circular motion. Once we solve the differential equation a = −ω²x (next lesson) we will find that one complete oscillation takes a time
T = 2π/ω and ω = 2πf
These are identical to the circular-motion relationships. You should commit them to memory — they appear in virtually every SHM calculation.
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