Preview
IB DP Physics — Unit C: Wave Behaviour | SL & HL | C.1 Simple Harmonic Motion
📋 What You Will Learn
- Observe oscillatory motion in real-world systems and identify what makes it periodic
- Define displacement, amplitude, period, frequency, and angular frequency with correct units and notation
- State and apply the defining condition for simple harmonic motion: \( a = -\omega^2 x \)
- Derive the relationship between period, frequency, and angular frequency from first principles
- Analyse displacement–time, velocity–time, and acceleration–displacement graphs for SHM
- Evaluate common misconceptions and connect SHM to energy, waves, and resonance across the IB Physics course
🌍 Real-World Context Watch a child on a playground swing, a guitar string vibrating after being plucked, or the needle of a sewing machine moving up and down. Each of these objects moves back and forth repeatedly, always returning to the same central position, always taking the same amount of time for each complete cycle. This kind of motion — rhythmic, repetitive, and symmetric — is one of the most fundamental patterns in all of physics, and it underlies everything from the ticking of a clock to the propagation of sound and light waves.
🔍 Guiding Question What is the precise mathematical condition that distinguishes simple harmonic motion from other types of oscillation, and how do the key quantities — displacement, amplitude, period, frequency, and angular frequency — describe the state of an oscillating system at any instant?
Physical Setup and Observation
Consider a small mass attached to a horizontal spring on a frictionless surface. When the mass is pulled to one side and released, it moves back toward its original resting position — but it does not stop there. It overshoots, compresses the spring on the other side, slows down, and then returns again. This back-and-forth motion continues indefinitely (in the absence of friction), with the mass passing through the centre every half-cycle.
What we observe is that the mass always turns around at the same distance from the centre, always takes the same time to complete one full cycle, and always accelerates most strongly when it is farthest from the centre. These three observations are the physical fingerprints of simple harmonic motion, and our task is to express them precisely in mathematical language.
Notice from the diagram that the restoring force is always directed toward the equilibrium position — it points left when the mass is to the right, and right when the mass is to the left. At the equilibrium position itself, the net force is zero, but the mass has its maximum speed. At the extreme positions (maximum displacement), the force is greatest but the speed is zero. This interplay between force and velocity is the essence of SHM.
Core Concepts and Definitions
Before we can write any equations, we need to establish a precise vocabulary. The mass in our spring system is not simply "moving back and forth" — it is oscillating, and each aspect of that oscillation has a specific, measurable quantity associated with it. We will build these definitions one at a time, in the order they are needed.
📖 Key Definition — Oscillation An oscillation is one complete back-and-forth motion of a system about its equilibrium position. The system returns to its starting position and starting velocity after one complete oscillation.
The first quantity we need is a way to describe where the object is at any given moment. We measure this as a signed distance from the equilibrium position —