Physics · Ch 10 — Oscillations
The projection of uniform circular motion on a diameter of SHM
The projection of uniform circular motion on a diameter of SHM
A neat geometric way to picture SHM is as the shadow of uniform circular motion. Consider a particle moving with constant speed on the circumference of a circle of radius , centred at the origin, in the anticlockwise sense. If is its constant angular velocity and its angular displacement at time , then . Now project this moving particle onto one diameter of the circle -- imagine shining a light from far away so the particle casts a shadow onto a screen along that diameter. As the particle goes round and round the circle at constant speed, its shadow on the diameter moves back and forth, executing exactly Simple Harmonic Motion. The circle used for this construction is called the reference circle of the SHM. From the right triangle formed by the particle's position, the origin, and the foot of the perpendicular on the diameter, (with the projected displacement and the radius), so -- exactly the displacement equation of SHM. The relationship works both ways: any SHM can be represented as the projection of some uniform circular motion, and conversely any uniform circu …
What this figure shows. A particle P moves anticlockwise at constant speed around a reference circle of radius A. A distant light source shines across the circle so that P casts a moving shadow onto a screen placed along the circle's vertical diameter; as P sweeps around at constant angular velocity omega, its shadow slides up and down the screen. The angle theta = omega t that OP makes with the horizontal axis is marked, together with the projected height y = A sin(theta) = A sin(omega t) and the horizontal companion x = A cos(theta), showing geometrically why a shadow of uniform circul …
What this figure shows. Eight snapshots (labelled P1 through P8) show the same particle at eight equally-spaced instants as it goes once around the reference circle, with a small marker on the vertical axis in each panel showing where the corresponding projected shadow point sits at that instant. Reading the eight panels in sequence shows the projected point sliding from the centre up to the top (maximum positive displacement), back down through the centre to the bottom (maximum negative displacement), and back up to the centre again -- one complete cycle of SHM traced out by one complete revolution of …
What this figure shows. A spring-mass system oscillating vertically is drawn side by side with a point moving around a reference circle, connected by dashed guide lines showing that the block's instantaneous height always matches the height of the corresponding point on the circle. The figure makes the abstract projection idea concrete: the up-and-down motion of a real spring-mass system (or equally, the to-and-fro swing of a pendulum bob) is physically identical, moment by moment, to the vertical shadow of a point moving steadily around the reference circle at the oscillator's own a …