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

Question

For a damped harmonic oscillation, the equation of motion is

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

$\text{with}m=0.25\text{kg},\gamma=0.05\text{kgs}^{-1}\text{and}k=70\text{Nm}^{-1}.$ 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.

30 May 2025

Answer :

Word Count : 309

We are solving numerically and manually the damped harmonic motion problem: $$ m\frac{d^2x}{dt^2}+\gamma \frac{dx}{dt}+kx=0 $$ with $$ m = 0.25 \text{ kg}, \quad \gamma = 0.05 \text{ kg/s}, \quad k = 70 \text{ N/m}. $$ We are to calculate: 1. Period of motion 2. Number of oscillations in which amplitude becomes half 3. Number of oscillations in which energy becomes half ________ ________ _________ ___ ___ ___ ___ _______ _____ ____ ______.
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