Physics · Ch 10 — Oscillations
Periodic and non-periodic motion
Periodic and non-periodic motion
Motion in physics is broadly classified as repetitive (periodic) or non-repetitive (non-periodic). Periodic motion is any motion that repeats itself after a fixed interval of time -- the hands of a pendulum clock returning to the same position, the swing of a cradle completing each cycle in the same time, the annual revolution of the Earth around the Sun, and the monthly waxing and waning of the Moon are all periodic. Non-periodic motion, in contrast, is motion that does not repeat itself after any regular time interval -- the sudden occurrence of an earthquake or the eruption of a volcano are examples; there is no fixed interval after which these events recur. The distinguishing test is simple: can a fixed duration be named such that the exact state of the system (position and velocity) recurs after every multiple of ? If yes, the motion is periodic; if the recurrence interval keeps changing, or the s …
Worked out. The question asks whether the motion of Halley's comet, the motion of clouds, and the Moon revolving around the Earth are periodic or non-periodic. Halley's comet returns to the inner solar system on a fixed, regularly repeating schedule (about every 76 years), so its orbital motion is periodic. The Moon revolving around the Earth likewise completes one orbit in a fixed, repeating interval (about 27.3 days), so it too is periodic motion. The motion of clouds, however, is governed by constantly changing wind patterns, temperature and pressure systems with no fixed repeating interval, so c …
Worked out. The question asks which of two functions of time represent periodic or non-periodic motion: (a) sin(omega t) + cos(omega t), and (b) ln(omega t). For (a), both sine and cosine are periodic functions with period 2 pi/omega, and their sum is also periodic with the same period, since the whole expression repeats identically every time t increases by 2 pi/omega; so this function represents periodic motion. For (b), the natural logarithm ln(omega t) grows without bound as t increases and never returns to a previous value at any later fixed interval, so it never repeats itself; this function is non-periodic. The general lesson is that any sum of sines and cosines of the same argument is periodic, while a mo …