Question

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 1 4 of this value. Find the damping constant c for this tuning fork. (Round your answer to two decimal places.)

Answer 1

Answer:

C=0.46 s⁻¹

Step-by-step explanation:

The general equation for the damped harmonic system given as

$y=Xe^{-Ct}sin\omega t$

Where

A= Amplitude

$A=Xe^{-Ct}$

C=Damping constant

Lets take at time t= t₁,  A= K

And at time t= t₁+3 s ,A= K/4

$A=Xe^{-Ct}$

$K=Xe^{-Ct_1}$                   -----------1

$\dfrac{K}{4}=Ke^{-C(t_1+3)}$                    -------------2

From equation 1 and 2

$Xe^{-Ct_1}=Ke^{-C(t_1+3)}$

$1=4e^{-3C}$

$e^{3C}=4$

$C=\dfrac{1}{3}\ln4$

C=0.46 s⁻¹