Question
For a damped harmonic oscillator, the equation of motion is
with m = 0.50 kg, = 0.70 kgs−1 and k = 70 Nm−1. Calculate (i) the period of motion, (ii) number of oscillations in which its amplitude will become half of its initial value, (iii) the number of oscillations in which its mechanical energy will drop to half of its initial value, (iv) its relaxation time, and (v) quality factor.
Answer :
Word Count : 457
To solve the given problem for the damped harmonic oscillator, we'll follow these steps: ### Given: - Mass, \( m = 0.50 \, \text{kg} \) - Damping coefficient, \( \gamma = 0.70 \, \text{kg/s} \) - Spring constant, \( k = 70 \, \text{N/m} \) ### (i) Period of Motion For a damped harmonic oscillator, the angular frequency \( \omega' \) is given by: \[ \omega' = \sqrt{\frac{k}{m} - \left(\frac{\gamma}{2m}\right)^2} \] First, calculate the natural angular _______ ____ _______ ___ _________ ______ _______ __________ _____ ____.
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To solve the given problem for the damped harmonic oscillator, we'll follow these steps: ### Given: - Mass, \( m = 0.50 \, \text{kg} \) - Damping coefficient, \( \gamma = 0.70 \, \text{kg/s} \) - Spring constant, \( k = 70 \, \text{N/m} \) ### (i) Period of Motion For a damped harmonic oscillator, the angular frequency \( \omega' \) is given by: \[ \omega' = \sqrt{\frac{k}{m} - \left(\frac{\gamma}{2m}\right)^2} \] First, calculate the natural angular _______ ____ _______ ___ _________ ______ _______ __________ _____ ____.
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