Precalculus: Mathematics for Calculus - 6th Edition
Precalculus: Mathematics for Calculus - 6th Edition
6th Edition
ISBN: 9780840068071
Author: Stewart, James, Redlin, Lothar, Watson, Saleem
Publisher: Cengage Learning
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Textbook Question
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Chapter 5.6, Problem 48E

Guitar String A guitar string is pulled at point P a distance of 3 cm above its rest position. It is then released and vibrates in damped harmonic motion with a frequency of 165 cycles per second. After 2 s, it is observed that the amplitude of the vibration at point P is 0.6 cm.

  1. (a) Find the damping constant c.
  2. (b) Find an equation that describes the position of point P above its rest position as a function of time. Take t = 0 to be the instant that the string is released.

(a)

Expert Solution
Check Mark
To determine

To find: The damping constant c for the guitar string which oscillates in damped harmonic motion.

Answer to Problem 48E

The damping constant c for the guitar string is 0.80s.

Explanation of Solution

Given:

The amplitude after 2s is 0.6 cm, frequency is 165cyclespersecond and the initial displacement of the guitar is 3 cm.

Definition used:

The equation for the damped harmonic motion which describes the displacement y of an object at time t is,

y=kectsinωt (1)

Calculation:

The general equation for the harmonic motion is,

y=asinωt (2)

Compare equation (1) and (2),

a=kect (3)

Assume t=0, when the tuning fork start oscillation. At t=0, ect=1.

Substitute 0 for t, and 1 for ect in equation (3).

Calculate the value of a at t=0.

a=kec0a=k

Calculate the value of a at t=2.

a=kec2a=ke2c

Since the value of amplitude after 2s is 0.6. So,

a(0)a(2)=30.6 (4)

Substitute k for a(0) and k2c for a(2) in equation (4).

kke2c=30.61e2c=5e2c=15

Take log on both the sides,

lne2c=ln152c=1.609c=0.80

Thus the value of damping coefficient for the tuning fork is 0.80.

(b)

Expert Solution
Check Mark
To determine

To find: The equation which models the displacement of the guitar string as a function of time.

Answer to Problem 48E

The equation for the displacement of the shock absorber as a function of time

is f(t)=3e0.8tcos330πt.

Explanation of Solution

Definition used:

The equation for the damped harmonic motion which describes the maximum displacement y of an object at time t is,

y=kectcosωt (5)

Calculation:

Calculate the equation for the displacement of the guitar string from its rest position as follows.

When the guitar string is released, take t=0.

The formula to calculate the value of ω is,

ω=2πf[f=1p]

Substitute the value 165 for f in ω=2πf.

ω=2π×165=330π

The value of ω is 330π.

Substitute the value 3 for k, 0.8 for c and 330π for ω in equation (5).

y=3e0.8tcos(330πt)=3e0.8tcos330πt

Hence, the function for the displacement is y=3e0.8tcos330πt.

Chapter 5 Solutions

Precalculus: Mathematics for Calculus - 6th Edition

Ch. 5.1 - Prob. 11ECh. 5.1 - Prob. 12ECh. 5.1 - Prob. 13ECh. 5.1 - Prob. 14ECh. 5.1 - Prob. 15ECh. 5.1 - Prob. 16ECh. 5.1 - Prob. 17ECh. 5.1 - Prob. 18ECh. 5.1 - Prob. 19ECh. 5.1 - Prob. 20ECh. 5.1 - Prob. 21ECh. 5.1 - Prob. 22ECh. 5.1 - Prob. 23ECh. 5.1 - Prob. 24ECh. 5.1 - Prob. 25ECh. 5.1 - Prob. 26ECh. 5.1 - Prob. 27ECh. 5.1 - Prob. 28ECh. 5.1 - Prob. 29ECh. 5.1 - Prob. 30ECh. 5.1 - Prob. 31ECh. 5.1 - Prob. 32ECh. 5.1 - Prob. 33ECh. 5.1 - Prob. 34ECh. 5.1 - Prob. 35ECh. 5.1 - Prob. 36ECh. 5.1 - Prob. 37ECh. 5.1 - Prob. 38ECh. 5.1 - Prob. 39ECh. 5.1 - Prob. 40ECh. 5.1 - Prob. 41ECh. 5.1 - Prob. 42ECh. 5.1 - Prob. 43ECh. 5.1 - Prob. 44ECh. 5.1 - Prob. 45ECh. 5.1 - Prob. 46ECh. 5.1 - Prob. 47ECh. 5.1 - Prob. 48ECh. 5.1 - Prob. 49ECh. 5.1 - Prob. 50ECh. 5.1 - Prob. 51ECh. 5.1 - Prob. 52ECh. 5.1 - Prob. 53ECh. 5.1 - Prob. 54ECh. 5.1 - Prob. 55ECh. 5.1 - Prob. 56ECh. 5.1 - Prob. 57ECh. 5.1 - Prob. 58ECh. 5.2 - Let P(x, y) be the terminal point on the unit...Ch. 5.2 - If P(x, y) is on the unit circle, then x2 + y2 =...Ch. 5.2 - 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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 - Prob. 1RCCCh. 5 - Prob. 2RCCCh. 5 - Prob. 3RCCCh. 5 - Prob. 4RCCCh. 5 - Prob. 5RCCCh. 5 - Prob. 6RCCCh. 5 - Prob. 7RCCCh. 5 - Prob. 8RCCCh. 5 - Prob. 9RCCCh. 5 - Prob. 10RCCCh. 5 - Prob. 11RCCCh. 5 - Prob. 12RCCCh. 5 - Prob. 1RECh. 5 - Prob. 2RECh. 5 - Prob. 3RECh. 5 - Prob. 4RECh. 5 - Prob. 5RECh. 5 - Prob. 6RECh. 5 - Prob. 7RECh. 5 - Prob. 8RECh. 5 - Prob. 9RECh. 5 - Prob. 10RECh. 5 - Prob. 11RECh. 5 - Prob. 12RECh. 5 - Prob. 13RECh. 5 - Prob. 14RECh. 5 - Prob. 15RECh. 5 - Prob. 16RECh. 5 - Prob. 17RECh. 5 - Prob. 18RECh. 5 - Prob. 19RECh. 5 - Prob. 20RECh. 5 - Prob. 21RECh. 5 - Prob. 22RECh. 5 - Prob. 23RECh. 5 - Prob. 24RECh. 5 - 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