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Simple Harmonic Motion A mass suspended from a spring oscillates in simple harmonic motion at a frequency of 4 cycles per second. The distance from the highest to the lowest point of the oscillation is 100 cm. Find an equation that describes the distance of the mass from its rest position as a function of time. Assume that the mass is at its lowest point when t = 0.

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Precalculus: Mathematics for Calcu...

7th Edition
James Stewart + 2 others
Publisher: Brooks Cole
ISBN: 9781305071759

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Section
BuyFindarrow_forward

Precalculus: Mathematics for Calcu...

7th Edition
James Stewart + 2 others
Publisher: Brooks Cole
ISBN: 9781305071759
Chapter 5, Problem 73RE
Textbook Problem
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Simple Harmonic Motion A mass suspended from a spring oscillates in simple harmonic motion at a frequency of 4 cycles per second. The distance from the highest to the lowest point of the oscillation is 100 cm. Find an equation that describes the distance of the mass from its rest position as a function of time. Assume that the mass is at its lowest point when t = 0.

To determine

The equation that describes the distance of the mass from its rest point as a function of time.

Explanation of Solution

Harmonic motion is a form of periodic motion in which an object oscillates in a friction less environment about a central point. The harmonic function can be defined with the use of sine and cosine functions.

Consider a sine and cosine equations,

y=asinωt (1)

y=acosωt (2)

Where,

|a| is the amplitude that is the maximum displacement.

2πω is the period that is the time required for the complete cycle.

These equation describes the displacement of an object from its equilibrium point at time t=0 .

It is given that the frequency of the harmonic motion is 4 cycles/s

ω2π=4cycles/sω=8π

And, the distance of the mass from the highest to the lowest point point of the oscillation is 100

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Chapter 5 Solutions

Precalculus: Mathematics for Calculus (Standalone Book)
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Variable Stars Astronomers believe that the radius...Ch. 5.6 - Biological Clocks Circadian rhythms are biological...Ch. 5.6 - Electric Generator The armature in an electric...Ch. 5.6 - Electric Generator The graph shows an oscilloscope...Ch. 5.6 - Doppler Effect When a car with its horn blowing...Ch. 5.6 - Motion of a Building A strong gust of wind strikes...Ch. 5.6 - Shock Absorber When a car hits a certain bump on...Ch. 5.6 - Tuning Fork A tuning fork is struck and oscillates...Ch. 5.6 - Guitar String A guitar string is pulled at point P...Ch. 5.6 - Two Fans Electric fans A and B have radius 1 ft...Ch. 5.6 - Alternating Current Alternating current is...Ch. 5.6 - DISCUSS: Phases of Sine The phase of a sine curve...Ch. 5.6 - DISCUSS: Phases of the Moon During the course of a...Ch. 5 - (a) What is the unit circle, and what is the...Ch. 5 - Let t be a real number, and let P(x, y) be the...Ch. 5 - (a) Describe the steps we use to find the value of...Ch. 5 - (a) What is a periodic function? (b) What are the...Ch. 5 - (a) What is an even function, and what is an odd...Ch. 5 - (a) State the reciprocal identities. (b) Slate the...Ch. 5 - (a) Graph the sine and cosine functions. (b) What...Ch. 5 - (a) Graph the tangent and cotangent functions. (b)...Ch. 5 - (a) Graph the cosecant and secant functions. (b)...Ch. 5 - (a) Define the inverse sine function, the inverse...Ch. 5 - (a) What is simple harmonic motion? (b) What is...Ch. 5 - Suppose that an object is in simple harmonic...Ch. 5 - Consider the following models of harmonic motion....Ch. 5 - Terminal Points A point P(x, y) is given. (a) Show...Ch. 5 - Terminal Points A point P(x, y) is given. (a) Show...Ch. 5 - Reference Number and Terminal Point A real number...Ch. 5 - Reference Number and Terminal Point A real number...Ch. 5 - Reference Number and Terminal Point A real number...Ch. 5 - Reference Number and Terminal Point A real number...Ch. 5 - Values of Trigonometric Functions Find the value...Ch. 5 - Values of Trigonometric Functions Find the value...Ch. 5 - Values of Trigonometric Functions Find the value...Ch. 5 - Values of Trigonometric Functions Find the value...Ch. 5 - Values of Trigonometric Functions Find the value...Ch. 5 - Values of Trigonometric Functions Find the value...Ch. 5 - Values of Trigonometric Functions Find the value...Ch. 5 - Values of Trigonometric Functions Find the value...Ch. 5 - Values of Trigonometric Functions Find the value...Ch. 5 - Values of Trigonometric Functions Find the value...Ch. 5 - Fundamental Identities Use the fundamental...Ch. 5 - Fundamental Identities Use the fundamental...Ch. 5 - Fundamental Identities Use the fundamental...Ch. 5 - Fundamental Identities Use the fundamental...Ch. 5 - Values of Trigonometric Functions Find the values...Ch. 5 - Values of Trigonometric Functions Find the values...Ch. 5 - Values of Trigonometric Functions Find the values...Ch. 5 - Values of Trigonometric Functions Find the values...Ch. 5 - Values of Trigonometric Functions Find the values...Ch. 5 - Values of Trigonometric Functions Find the values...Ch. 5 - Values of Trigonometric Functions Find the values...Ch. 5 - Values of Trigonometric Functions Find the values...Ch. 5 - Horizontal Shifts A trigonometric function is...Ch. 5 - Horizontal Shifts A trigonometric function is...Ch. 5 - Horizontal Shifts A trigonometric function is...Ch. 5 - Horizontal Shifts A trigonometric function is...Ch. 5 - Horizontal Shifts A trigonometric function is...Ch. 5 - Horizontal Shifts A trigonometric function is...Ch. 5 - Horizontal Shifts A trigonometric function is...Ch. 5 - Horizontal Shifts A trigonometric function is...Ch. 5 - Functions from a Graph The graph of one period of...Ch. 5 - Functions from a Graph The graph of one period of...Ch. 5 - Functions from a Graph The graph of one period of...Ch. 5 - Functions from a Graph The graph of one period of...Ch. 5 - Graphing Trigonometric Functions Find the period,...Ch. 5 - Graphing Trigonometric Functions Find the period,...Ch. 5 - Graphing Trigonometric Functions Find the period,...Ch. 5 - Graphing Trigonometric Functions Find the period,...Ch. 5 - Graphing Trigonometric Functions Find the period,...Ch. 5 - Graphing Trigonometric Functions Find the period,...Ch. 5 - Graphing Trigonometric Functions Find the period,...Ch. 5 - Graphing Trigonometric Functions Find the period,...Ch. 5 - Evaluating Expressions Involving Inverse...Ch. 5 - Evaluating Expressions Involving Inverse...Ch. 5 - Evaluating Expressions Involving Inverse...Ch. 5 - Evaluating Expressions Involving Inverse...Ch. 5 - Amplitude, Period, Phase, and Horizontal Shift For...Ch. 5 - Amplitude, Period, Phase, and Horizontal Shift For...Ch. 5 - Phase and Phase Difference A pair of sine curves...Ch. 5 - Phase and Phase Difference A pair of sine curves...Ch. 5 - Even and Odd Functions A function is given. (a)...Ch. 5 - Even and Odd Functions A function is given. (a)...Ch. 5 - Even and Odd Functions A function is given. (a)...Ch. 5 - Even and Odd Functions A function is given. (a)...Ch. 5 - Even and Odd Functions A function is given. (a)...Ch. 5 - Even and Odd Functions A function is given. (a)...Ch. 5 - Sine and Cosine Curves with Variable Amplitude...Ch. 5 - Sine and Cosine Curves with Variable Amplitude...Ch. 5 - Sine and Cosine Curves with Variable Amplitude...Ch. 5 - Sine and Cosine Curves with Variable Amplitude...Ch. 5 - Maxima and Minima Find the maximum and minimum...Ch. 5 - Maxima and Minima Find the maximum and minimum...Ch. 5 - Solving Trigonometric Equations Graphically Find...Ch. 5 - Solving Trigonometric Equations Graphically Find...Ch. 5 - Discover the Period of a Trigonometric Function...Ch. 5 - Simple Harmonic Motion A point P moving in simple...Ch. 5 - Simple Harmonic Motion A mass suspended from a...Ch. 5 - Damped Harmonic Motion The top floor of a building...Ch. 5 - The point P(x, y) is on the unit circle in...Ch. 5 - The point P in the figure at the left has...Ch. 5 - Find the exact value. (a) sin76 (b) cos134 (c)...Ch. 5 - Express tan t in terms of sin t, if the terminal...Ch. 5 - If cost=817 and if the terminal point determined...Ch. 5 - A trigonometric function is given. (a) Find the...Ch. 5 - A trigonometric function is given. (a) Find the...Ch. 5 - Find the period, and graph the function. 8. y =...Ch. 5 - Find the period, and graph the function. 9....Ch. 5 - Find the exact value of each expression, if it is...Ch. 5 - The graph shown at left is one period of a...Ch. 5 - The sine curves y1=30sin(6t2) and y2=30sin(6t3)...Ch. 5 - Let f(x)=cosx1+x2. (a) Use a graphing device to...Ch. 5 - A mass suspended from a spring oscillates in...Ch. 5 - An object is moving up and down in damped harmonic...

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