Question

In a damped oscillator with m = 500 g, k = 100 N/m, and b = 75 g/s, the ratio of the amplitude of the damped oscillations to the initial amplitude at the end of 20 cycles is given as $e^{-x}$. Find $x$.

• A. 0.555
• B. 0.666
• C. 0.777
• D. 0.888

Answer 1

The correct option is B 0.666

The time period of this system is given by
$T=2\pi \sqrt{\frac{m}{k}}=2\pi \sqrt{\frac{0.5}{100}}=\frac{\pi }{5\sqrt{2}}sec$
For 20 cycles, time taken will be $=t=20\times \frac{\pi }{5\sqrt{2}}=2\sqrt{2}\pi sec$
So the ratio of amplitude after 20 cycles to that of initial amplitude is
$\frac{y}{y_o}=e^{-bt}=e^{(-\frac{75\times 2\sqrt{2}\pi }{1000})}=e^{-0.666}$