Question

Stringed instruments like guitars and pianos create a note when a string vibrates back and forth. The distance that the middle of the string moves from the center is called the amplitude (a), and for a guitar, it starts at 0.75 mm when a note is first struck. Amplitude decays at a rate that depends on the individual instrument and the note, but a decay rate of about 25% per second is typical. Calculate the time it takes for an amplitude of 0.75 mm to reach 0.1 mm.
The amplitude will reach 0.1 mm in about 7 seconds.
The amplitude will reach 0.1 mm in about 7 seconds.

The amplitude will reach 0.1 mm in about 15 seconds.
The amplitude will reach 0.1 mm in about 15 seconds.

The amplitude will reach 0.1 mm in about 1.5 seconds.
The amplitude will reach 0.1 mm in about 1.5 seconds.

The amplitude will reach 0.1 mm in about 3 seconds.

Answer 1

The the product of the initial amplitude and the exponential decay rate

of raised to the power of the time taken give the amplitude of 0.1 mm.

Correct option:

• The amplitude will reach 0.1 mm in about 7 seconds

What is the exponential decay rate and how is it used?

The given parameters are;

The initial amplitude = 0.75 mm

The decay rate, i = 25%

Required:

The time it takes the guitar to decay to 0.1 mm.

Solution:

The series of the sizes of the amplitude that has constant decay uses an

exponential decay formula.

When the amplitude is 0.1 mm, we have;

y = a·rⁿ

Where;

r = 1 - i

a = 0.75

n = The time taken

Which gives;

r = 1 - 0.25 = 0.75

y = 0.1 = 0.75 × 0.75ⁿ

$\dfrac{0.1}{0.75} = \mathbf{0.75^{n}}$

• $n = \dfrac{ln\left( \dfrac{0.1}{0.75} \right)}{ln(0.75)} \approx \mathbf{7.0}$

n = 7.0

• The time it takes the amplitude to reach 0.1 mm is 7 seconds.

Learn more about exponential growth and decay formula here:

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