Chapter 5.6, Problem 47E

Tuning Fork A tuning fork is struck and oscillates in damped harmonic motion. The amplitude of the motion is measured, and 3 s later it is found that the amplitude has dropped to $\frac{1}{4}$ of this value. Find the damping constant c for this tuning fork.

To determine

The damping coefficient c for this tuning fork.

Answer to Problem 47E

The damping coefficient for the tuning fork is $0.46$.

Explanation of Solution

Given:

In time $3\text{s}$ the amplitude reaches $\frac{1}{4}$ of its initial value.

Definition used:

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

$y=k{e}^{-ct}\sin \omega t$ (1)

Calculation:

The general equation for the harmonic motion is,

$y=a\sin \omega t$ (2)

Compare equation (1) and (2),

$a=k{e}^{-ct}$ (3)

The amplitude dropped $\frac{1}{4}$ of its initial value in time $3\text{s}$.

Assume $t=0$, when the tuning fork start oscillation. At $t=0$, $e^{-ct}=1$.

Substitute 0 for t, and 1 for $e^{-ct}$ in equation (3),

Calculate the value of a at $t=0$,

$\begin{array}{l}a=k{e}^{-c\cdot 0}\\ a=k\end{array}$

Calculate the value of a at $t=3$,

$\begin{array}{l}a=k{e}^{-c\cdot 3}\\ a=k{e}^{-3c}\end{array}$

Since the value of amplitude is dropped to $\frac{1}{4}$ of its initial value.

So,

$\frac{a\left(0\right)}{a\left(3\right)}=4$ (4)

Substitute k for $a\left(0\right)$ and $k^{-3c}$ for $a\left(3\right)$ in equation (4),

$\begin{array}{c}\frac{k}{k{e}^{-3c}}=4\\ \frac{1}{e^{-3c}}=4\\ e^{-3c}=\frac{1}{4}\end{array}$

Take log on both the sides,

$\begin{array}{c}\ln e^{-3c}=\ln \frac{1}{4}\\ -3c=-1.386\\ c=0.46\end{array}$

Thus, the value of damping coefficient for the tuning fork is $0.46$.