The correct option is B 0.344
Let us consider two oscillations , one occuring ′m′ cycles after the first oscillation.
Amplitude of first oscillation at time t is given by x1=Ae−δω0t .....(1)
where δ=γω0
Amplitude of second oscillation after m cycles (time mT) is given by x2=Ae−δω0(t+mT) ......(2)
From (1) and (2), we can write that,
x1x2=eδω0mT
⇒δω0mT=ln(x1x2) ......(3)
Here, δ is known as damping ratio.
For successive amplitudes, m=1
∴δω0T=ln(101)=2.3
⇒δω0(2πω′)=2.3 ....(4)
But we know that, ω′=√ω20−γ2
From this we get, ω′=ω0√1−δ2
where δ=γω0
From (4) we can write that,
2πδ√1−δ2=2.3
Squaring on both sides ,
39.4 δ2(1−δ2)=5.29
⇒δ2=0.118⇒δ=0.344
Thus, option (b) is the correct answer.