Example 13 from Precalculus, 7.6 Modeling with Trigonometric Functions

Finding the Value of the Damping Constant c According to the Given Criteria

A guitar string is plucked and vibrates in damped harmonic motion. The string is pulled and displaced 2 cm from its resting position. After 3 seconds, the displacement of the string measures 1 cm. Find the damping constant.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
The displacement factor represents the amplitude and is determined by the coefficient $a e^{- c t}$ in the model for damped harmonic motion. The damping constant is included in the term $e^{- c t} .$ It is known that after 3 seconds, the local maximum measures one-half of its original value. Therefore, we have the equation

$a e^{- c (t + 3)} = \frac{1}{2} a e^{- c t}$
Use algebra and the laws of exponents to solve for $c .$

$\begin{array}{l} a e^{- c (t + 3)} = \frac{1}{2} a e^{- c t} \\ e^{- c t} \cdot e^{- 3 c} = \frac{1}{2} e^{- c t} \begin{matrix} \end{matrix} & Divide out a . \\ e^{- 3 c} = \frac{1}{2} & Divide out e^{- c t} . \\ e^{3 c} = 2 & Take reciprocals . \end{array}$
Then use the laws of logarithms.

$\begin{array}{l} e^{3 c} = 2 \\ 3 c = \ln (2) \\ c = \frac{\ln (2)}{3} \end{array}$
The damping constant is $\frac{\ln (2)}{3} .$

How did it go?

Solution

Solution