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) $$ \omega_0 = \sqrt{\frac{k}{m}} = \sqrt{\frac{70}{0.25}} = \sqrt{280} \approx 16.733 \, \text{rad/s} $$ --- ### Step 2: Damping coefficient per unit mass $$ \frac{\gamma}{2m} = \frac{0.05}{2 \times 0.25} = \frac{0.05}{0.5} = 0.1 \, \text{s}^{-1} $$ _________ _________ _____ _________ ______ ______.
_______ ____ _________ _______ ___ _________ _________ _______ ________.
____ _____ _________ _______ _______ __________ ___ _____ ______ _________ ___.
______ _________ _________ _______ _____.
______ ______ __________ ___ ________ ___ ______ ______ _________ ____ ______ ____.
___ ______ ____ _____ _______ __________ ____ _______ _______.
__________ _____ ________ ____ _____ _________.
__________ _____ _________ _________ _________.
_______ __________ _____ _________ __________ _____ ______ _________.
____ ______ ___ _______ ___ _________ ________ _______ _____ ____.
____ _________ _______ __________ ____ _________ _________ _____.
________ _______ _______ _______ _______ __________ __________ _________ ____ ___.
_______ _____ __________ ________ _______ _________.
_________ ________ _______ ________ ____ ___ ______ ______ ______ ________.
_________ __________ ______ __________ _____ __________ _______ ____ ___.
________ ________ _____ __________ ________.
____ ____ _______ _________ ______ ____ ______ _________ _______ _____ _______.
_______ _____ ________ __________ _________ _______ _______ ____ ___ _________ ________ _______.
_______ ___ ____ _____ ___ ______.
_______ ___ _______ ___ __________ ____ _______.
__________ ______ ___ _________ ________ _________ ___ ______ _____ __________.
___ _______ ________ _____ ______ ___.
___ __________ ______ _________ ________ __________.
__________ ______ ___ __________ __________ ________ _____ _________ ______ _______.
_______ _______ _________ _______ ____ ___ ______ ____ _________ ________.
______ _____ _____ _____ ____ _______ ______ _____ _________ ____ __________ _____.
_____ ___ ______ _______ ______ _____ _______ _____ ___ _________ __________.
___ _________ ____ __________ _______ _____ _____ ____.
__________ _______.
Get Full Answer on WhatsApp

IGNOU Solved Assignments Assignments Ignou Projects Project Previous Year Solved Exam Papers Notes Notes Sample Question Papers Sample Question Papers

IGNOU NEWS

Assignment Submission Last Date Extended Till 30 June 2026 Click Here★★★IGNOU June 2026 TEE Date Sheet Released Click Here★★★