Harmonic Motion The displacement from equilibrium of an...

Harmonic Motion The displacement from equilibrium of an oscillating mass attached to a spring is given by $y(t)=4 \cos 3 \pi t$ where $y$ is measured in inches and $t$ in seconds. Find the displacement at the times indicated in the table.

Key Concepts

Period of Oscillation

The period is the time it takes for one complete cycle of the oscillation. It is inversely related to the frequency and can be calculated by dividing 2? by the angular frequency. This concept is essential for understanding the timing and repetition of the motion described by the displacement function.

Angular Frequency

Angular frequency measures how rapidly the object oscillates and appears as the coefficient of the time variable within the trigonometric function. It is directly related to the frequency and period of the oscillation, with higher angular frequency indicating more rapid cycles of motion.

Simple Harmonic Motion

This concept refers to a type of periodic motion in which the restoring force acting on an object is directly proportional to its displacement from equilibrium and acts in the opposite direction. It is a foundational idea in physics that explains many oscillatory systems, such as masses on springs and pendulums, through simple mathematical functions.

Displacement Function

A displacement function mathematically describes how far an oscillating object is from its equilibrium position at any given time. Typically expressed in terms of trigonometric functions like cosine or sine, these functions capture the periodic nature of the motion, including the effects of amplitude, frequency, and phase shift.

Amplitude

Amplitude is the maximum displacement of the oscillating object from its equilibrium position. It represents the energy and extent of the oscillatory motion and is indicated by the coefficient multiplying the trigonometric function in the displacement equation.

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