Problem 68 - Precalculus Essentials

Question

Explain the period and the frequency of simple harmonic motion. How are they related?

Step-by-Step Solution

Answer

In simple harmonic motion, the period is the time one full oscillation or cycle takes, while frequency denotes how many cycles are completed in a unit of time. They are inversely proportional to each other, as expressed by the formulas \(f = 1/T\) or \(T = 1/f\).

Step 1: Definition of Period in SHM

In Simple Harmonic Motion, the period (usually denoted as T) refers to the time it takes for an oscillating object to complete one full cycle or oscillation. This could be thought of as the time between two similar points in consecutive motions. For instance, the time it takes a swinging pendulum to swing from its farthest point on one end, to the farthest point on the other end, and then back to the original position.

Step 2: Definition of Frequency in SHM

Frequency (usually denoted as f or \(\nu\)) in Simple Harmonic Motion refers to the number of cycles or oscillations which an object completes in one unit of time, usually a second. It's the reciprocal of the period and, is usually measured in 'Hertz'(Hz), with one Hz equaling one oscillation per second. The higher the frequency, the more cycles or oscillations that are completed in a set frame of time and thus the quicker the motion.

Step 3: Relationship between Period and Frequency in SHM

In Simple Harmonic Motion, the relation between period and frequency is one of reciprocals - they are inversely proportional to each other. This means that as the period for one complete cycle increases, the frequency of oscillation decreases and vice versa. The period \(T\) and frequency \(f\) are related by the formula: \(f = 1/T\) or alternatively \(T = 1/f\). This relationship demonstrates that if you know the period, you can determine the frequency, and vice versa.