First slide

Damped and forced oscillations

Question

When a tuning fork (frequency 262 Hz) is struck, it loses half of its energy after 4s.The decay time ( $τ$ ) and Q-factor for this tuning fork respectively are (Given, ln 2 = 0.693)

Difficult

A

$4.77 s, 10.5 \times 10^{3}$
B

$5.77 s, 9.5 \times 10^{3}$
C

$6.77 s, 11.5 \times 10^{3}$
D

$2.77 s, 8.5 \times 10^{3}$

Solution

$W e k n o w E = E_{0} e^{- t / τ}$
$\begin{array}{l} E = \frac{E_{0}}{2} \\ ∴ \frac{E_{0}}{2} = E_{0} e^{- t / τ} \\ \Rightarrow e^{4 / τ} = 2 \end{array}$
Taking natural logarithms on both the sides
$\frac{4}{τ} = \ln 2$
$\Rightarrow τ = \frac{4}{\ln 2} = 5.77 s$
$N o w, Q = ω_{0} τ = (2 π f) τ = 2 \times 3.14 \times 262 \times 5.77 = 9.5 \times 10^{3}$

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