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
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.
- (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.
- (b) How long does it take for the amplitude of the vibration to decrease to 0.5 in.?
(a)
Expert Solution
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
Explanation of Solution
Given:
The frequency of the shock absorber is
Definition used:
The equation for the damped harmonic motion which describes the displacement y of an object at time t is,
Calculation:
Calculate the equation for the displacement of the shock absorber from it’s rests position.
Assume
The formula to calculate the value of
Substitute the value 2 for f in the above formula.
The value of
Substitute the value 6 for k,
Hence, the function for the displacement is
(b)
Expert Solution
To determine
The time for the amplitude of the vibration to decrease to
Answer to Problem 46E
The time for the amplitude of the vibration to decrease to
Explanation of Solution
Given:
The damping constant c is
Definition used:
The equation for the damped harmonic motion which describes the displacement y of an object at time t is,
Calculation:
Calculate the time for the amplitude of the vibration to decrease to 0.5 in.
The general equation for the harmonic motion is,
Compare equation (1) and (2),
Substitute the value 6 for k,
Further solve the value of equation,
Solve the value of t,
Thus, the value of the time for the amplitude is
Chapter 5 Solutions
Precalculus: Mathematics for Calculus - 6th Edition
Ch. 5.1 - Prob. 1ECh. 5.1 - Prob. 2ECh. 5.1 - Prob. 3ECh. 5.1 - Prob. 4ECh. 5.1 - Prob. 5ECh. 5.1 - Prob. 6ECh. 5.1 - Prob. 7ECh. 5.1 - Prob. 8ECh. 5.1 - Prob. 9ECh. 5.1 - Prob. 10E
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