Problem 2 - Precalculus with Limits

Question

A point that moves on a coordinate line is said to be in simple _____ _____ when its distance \(d\) from the origin at time \(t\) is given by either \(d=a \sin \omega t\) or \(d=a \cos \omega t\).

Step-by-Step Solution

Answer

The point moving on a coordinate line per the given equations is in simple harmonic motion.

Step 1: Understand Simple Harmonic Motion

In physics, simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement. It occurs when the object is displaced from its equilibrium position.

Step 2: Identify the properties of equations

This situation is represented by the equations \(d=a \sin \omega t\) or \(d=a \cos \omega t\). Here, \(a\) represents the amplitude or maximum displacement, \(\omega\) represents the angular frequency of the motion, and \(t\) represents time.

Step 3: Compare the equations with the definition

Comparing these two motion equations to the definition, we see that they both involve a sine and a cosine function, which are characteristic for simple harmonic motion. The distance \(d\) from the origin varies over time \(t\) with the frequency \(\omega\) and amplitude \(a\), providing evidence that this is indeed simple harmonic motion.