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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Chapter 5.6, Problem 46E

Shock Absorber When a car hits a certain bump on the road, a shock absorber on the car is compressed a distance of 6 in., then released (see the figure). The shock absorber vibrates in damped harmonic motion with a frequency of 2 cycles per second. The damping constant for this particular shock absorber is 2.8.

  1. (a) Find an equation that describes the displacement of the shock absorber from its rest position as a function of time. Take t = 0 to be the instant that the shock absorber is released.
  2. (b) How long does it take for the amplitude of the vibration to decrease to 0.5 in.?

Chapter 5.6, Problem 46E, Shock Absorber When a car hits a certain bump on the road, a shock absorber on the car is compressed

(a)

Expert Solution
Check Mark
To determine

The equation which models the displacement of the shock absorber as a function of time.

Answer to Problem 46E

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

is f(t)=6e−2.8tsin4πt .

Explanation of Solution

Given:

The frequency of the shock absorber is 2 cycle per second and the damping constant c is 2.8 . The initial amplitude of shock absorber is k is 6.

Definition used:

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

y=ke−ctsinωt (1)

Calculation:

Calculate the equation for the displacement of the shock absorber from it’s rests position.

Assume t=0 , when the shock absorber is released.

The formula to calculate the value of ω ,

ω=2πf                  [∴f=1p]

Substitute the value 2 for f in the above formula.

ω=2π×2=4π

The value of ω is 4π .

Substitute the value 6 for k, 2.8 for c and 4π for ω in equation (1),

y=6⋅e−2.8tsin(4πt)=6e−2.8tsin4πt

Hence, the function for the displacement is y=6e−2.8tsin4πt .

(b)

Expert Solution
Check Mark
To determine

The time for the amplitude of the vibration to decrease to 0.5 in.

Answer to Problem 46E

The time for the amplitude of the vibration to decrease to 0.5 in. is 0.89 s .

Explanation of Solution

Given:

The damping constant c is 2.8 . The initial amplitude of shock absorber is k is 6.

Definition used:

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

y=ke−ctsinωt (1)

Calculation:

Calculate the time for the amplitude of the vibration to decrease to 0.5 in.

The general equation for the harmonic motion is,

y=asinωt (2)

Compare equation (1) and (2),

a=ke−ct (3)

Substitute the value 6 for k, 2.8 for c and 0.5 for a in equation (3),

0.5=6e−2.8t0.56=e−2.8t0.0833=e−2.8t

Further solve the value of equation,

ln0.0833=−2.8t−2.485=−2.8t2.4852.8=t

Solve the value of t,

t=0.887st≈0.89 s

Thus, the value of the time for the amplitude is 0.89 s .

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Chapter 5.6, Problem 45E
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Chapter 5.6, Problem 47E

Chapter 5 Solutions

Precalculus: Mathematics for Calculus - 6th Edition

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