Understanding Angular Frequency and Period in Simple Harmonic Motion

Simple harmonic motion (SHM) is a fundamental concept in physics, describing the oscillations of a mass-spring system. This article will explore the concepts of angular frequency and period through a practical example involving a mass-spring system.

Introduction to Simple Harmonic Motion

In a mass-spring system, a 0.5 kg mass is attached to a spring, completing 13 oscillations in 20 seconds with an amplitude of 0.3 meters. The primary goal is to understand the relationship between the frequency of oscillation, the period of oscillation, and the angular frequency. This information will help us analyze the motion of the mass-spring system.

Finding the Frequency

The frequency ( f ) is defined as the number of oscillations or cycles per second. It is given by the formula:

[ f frac{1}{T} ]

Where ( T ) is the period of oscillation, the time taken for one complete cycle. Given that the system completes 13 oscillations in 20 seconds:

[ f frac{13}{20 , text{sec}} 0.65 , text{Hz} ]

Angular Frequency

The angular frequency ( omega ) is the measure of the frequency per unit time in radians. It is related to the frequency ( f ) by the formula:

[ omega 2pi f 2pi / T ]

Substituting the values:

[ omega 2pi times 0.65 , text{Hz} frac{2pi}{20 , text{sec}} 4.1 , text{rad/s} ]

Determining the Period

The period ( T ) can be directly calculated from the number of oscillations and the total time taken. Given that the system completes 13 oscillations in 20 seconds, the period is:

[ T frac{20 , text{sec}}{13} approx 1.54 , text{sec} ]

Which, to two significant figures, is approximately 1.5 seconds.

Conclusion and Practical Application

The period ( T ) is a crucial parameter in the study of SHM, as it directly relates to the time taken for one oscillation. The given information about the mass and amplitude of the oscillation is not necessary for determining the period in this specific scenario. The period is found using the simple relationship between the total time and the number of oscillations.