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Chapter 5: Problem 68

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

Short Answer

Expert verified
In Simple Harmonic Motion, the period is the time taken to complete one full oscillation, while frequency represents the number of oscillations per unit time. They are inversely related, with their relationship described by the equation \(f = 1 / T\). Therefore, a high frequency corresponds to a short period, and vice versa.

Step by step solution

01

Definition of Simple Harmonic Motion

Simple Harmonic Motion (SHM) is a type of oscillatory motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. Examples include pendulum of a clock, motion of a mass-spring system, etc.
02

Definition of Period

The period of motion, denoted as \(T\), is the time taken by the object to complete one full oscillation or cycle. It's calculated as the reciprocal of the frequency.
03

Definition of Frequency

The frequency of oscillations, denoted by \(f\), is the number of oscillations per unit time. It's the reciprocal of the period and is measured in Hertz (Hz).
04

Relationship between Period and Frequency

The period and frequency are inversely related. The mathematical relationship between the two is given by the formula \(f = 1 / T\). This means if the period is long (the motion is slow), the frequency will be low and vice versa. If the period is short (the motion is fast), the frequency will be high.
05

Example Illustrating The Relationship

Consider an object oscillating with a period of 0.2 s. According to the formula \(f = 1 / T\), we substitute \(T = 0.2 \text{s}\) and get a frequency \(f = 1 / 0.2 \text{Hz} = 5 \text{Hz}\). This means that in one second, 5 cycles of oscillation are completed by the object.

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Most popular questions from this chapter

Graph: \(f(x)=\frac{5 x+1}{x-1}\) (Section \(3.5, \text { Example } 5)\)

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