For a damped harmonic oscillation, the equation of motion is&n

Question

For a damped harmonic oscillation, the equation of motion is

$m\frac{d^2x}{dt^2}+\gamma\frac{dx}{dt}+kx=0$

$\text{with}m=0.25\,\text{kg},\gamma=0.05\,\text{kg s}^{-1}\text{and}k=70\,\text{N m}^{-1}.$ Calculate (i) the period of motion, (ii) number of oscillations in which its amplitude will become half of its initial value, and (iii) the number of oscillations in which its mechanical energy will drop to half of its initial value.

17 Apr 2025

Answer :

Word Count : 319

We are given the damped harmonic oscillator equation: $$ m\frac{d^2x}{dt^2} + \gamma\frac{dx}{dt} + kx = 0 $$ With: * $m = 0.25 \, \text{kg}$ * $\gamma = 0.05 \, \text{kg/s}$ * $k = 70 \, \text{N/m}$ --- ### Step 1: Natural angular frequency (undamped) $$ \omega_0 = \sqrt{\frac{k}{m}} = \sqrt{\frac{70}{0.25}} = \sqrt{280} \approx __________ ________ _______ _______ ___ _______ ________.
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