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.
- (a) Find the damping constant c.
- (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)
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
Explanation of Solution
Given:
The amplitude after
Definition used:
The equation for the damped harmonic motion which describes the displacement y of an object at time t is,
Calculation:
The general equation for the harmonic motion is,
Compare equation (1) and (2),
Assume
Substitute 0 for t, and 1 for
Calculate the value of a at
Calculate the value of a at
Since the value of amplitude after
Substitute k for
Take log on both the sides,
Thus the value of damping coefficient for the tuning fork is
(b)
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
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,
Calculation:
Calculate the equation for the displacement of the guitar string from its rest position as follows.
When the guitar string is released, take
The formula to calculate the value of
Substitute the value 165 for f in
The value of
Substitute the value 3 for k,
Hence, the function for the displacement is
Chapter 5 Solutions
Precalculus: Mathematics for Calculus - 6th Edition
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