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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 frequency \( \omega_0 \): \[ \omega_0 = \sqrt{\frac{k}{m}} = \sqrt{\frac{70}{0.50}} = \sqrt{140} \approx 11.83 \, \text{rad/s} \] Next, calculate the damping term: \[ \left(\frac{\gamma}{2m}\right)^2 = \left(\frac{0.70}{2 \times 0.50}\right)^2 = \left(\frac{0.70}{1.00}\right)^2 = 0.49 \, \text{rad}^2/\text{s}^2 \] Now, calculate \( _____ ____ ____ ___ ___ ___.
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