Student Details







Oscillations

The Damped Harmonic Oscillator

In a real-world spring or pendulum system, there is always some friction or resistance that removes energy from the system. This resistance is called damping.

1. Equation of Motion

Consider a mass \( m \) attached to a spring with constant \( k \) and a damping force proportional to velocity \( \dot{x} \): \[ F_d = -c\dot{x} \] where \( c \) is the damping coefficient.

The total force acting on the mass is: \[ F = -kx - c\dot{x} \] Applying Newton’s second law: \[ m\ddot{x} = -kx - c\dot{x} \quad \Rightarrow \quad m\ddot{x} + c\dot{x} + kx = 0 \]

This is the damped harmonic oscillator equation.

2. Standard Form

Dividing through by \( m \): \[ \ddot{x} + \frac{c}{m}\dot{x} + \frac{k}{m}x = 0 \] Let: \[ 2\beta = \frac{c}{m}, \quad \omega_0^2 = \frac{k}{m} \] Then the equation becomes: \[ \boxed{\ddot{x} + 2\beta\dot{x} + \omega_0^2 x = 0} \] where:

  • \( \omega_0 \) — natural (undamped) angular frequency
  • \( \beta \) — damping constant (\( \beta = c/2m \))

3. Nature of the Motion

The characteristic equation is: \[ r^2 + 2\beta r + \omega_0^2 = 0 \quad \Rightarrow \quad r = -\beta \pm \sqrt{\beta^2 - \omega_0^2} \] The behavior of the system depends on the relative size of \( \beta \) and \( \omega_0 \).

Case 1: Underdamped Motion (\( \beta < \omega_0 \))

Roots are complex: \[ r = -\beta \pm i\omega_d \quad \text{where} \quad \omega_d = \sqrt{\omega_0^2 - \beta^2} \] The solution is oscillatory: \[ x(t) = A e^{-\beta t}\cos(\omega_d t + \phi) \]

  • The amplitude decays exponentially with time.
  • The motion is sinusoidal, with reduced frequency \( \omega_d \).
  • Energy decreases as \( E \propto e^{-2\beta t} \).

Example: A pendulum oscillating slowly in air.

Case 2: Critically Damped Motion (\( \beta = \omega_0 \))

Roots are real and equal: \[ r_1 = r_2 = -\beta \] The general solution is: \[ x(t) = (A + Bt)e^{-\beta t} \]

  • No oscillations — the system returns to equilibrium without overshooting.
  • This is the fastest possible return to equilibrium.
  • Used in car shock absorbers and door closers.

Case 3: Overdamped Motion (\( \beta > \omega_0 \))

Roots are real and distinct: \[ r_{1,2} = -\beta \pm \sqrt{\beta^2 - \omega_0^2} \] The solution is: \[ x(t) = A e^{r_1 t} + B e^{r_2 t} \]

  • No oscillations — both exponential terms decay.
  • The system returns to equilibrium slowly compared to critical damping.

4. Energy in Damped Oscillations

The total mechanical energy decreases exponentially with time: \[ E(t) = E_0 e^{-2\beta t} \] where \( E_0 \) is the initial energy.

5. Comparison Summary

Type Condition Nature of Motion Equation of Motion
Underdamped \( \beta < \omega_0 \) Oscillatory with decaying amplitude \( x = A e^{-\beta t}\cos(\omega_d t + \phi) \)
Critically Damped \( \beta = \omega_0 \) Non-oscillatory, fastest return \( x = (A + Bt)e^{-\beta t} \)
Overdamped \( \beta > \omega_0 \) Non-oscillatory, slow return \( x = A e^{r_1 t} + B e^{r_2 t} \)

6. Graphical Behavior

  • Underdamped: Oscillations with gradually decreasing amplitude.
  • Critically Damped: Smooth exponential decay, no oscillation.
  • Overdamped: Even slower exponential decay, no oscillation.

Q1. The general equation of a damped harmonic oscillator is:




Q2. The damping force is proportional to:




Q3. In the standard form \( \ddot{x} + 2\beta\dot{x} + \omega_0^2 x = 0 \), the term \( 2\beta \) represents:




Q4. The solution to the damped oscillator equation depends on:




Q5. The characteristic equation of the damped harmonic oscillator is:




Q6. For underdamped motion, the condition is:




Q7. For critical damping, the system returns to equilibrium:




Q8. The motion is non-oscillatory when:




Q9. For underdamped motion, the displacement is given by:




Q10. The damped angular frequency \( \omega_d \) is related to \( \omega_0 \) and \( \beta \) as:




Q11. In an underdamped oscillator, the amplitude decreases:




Q12. The energy of a damped oscillator decays as:




Q13. In a critically damped system, the displacement is given by:




Q14. Which of the following systems should be critically damped?




Q15. For an overdamped system, the solution is:




Q16. In an overdamped oscillator, the system:




Q17. The critical damping coefficient \( c_c \) is given by:




Q18. The time taken for the amplitude to drop to \( 1/e \) of its original value is:




Q19. The quality factor \( Q \) of a lightly damped oscillator is:




Q20. A larger quality factor \( Q \) indicates:




Q21. The logarithmic decrement \( \delta \) is related to damping constant \( \beta \) as:




Q22. When damping is very small, the frequency of oscillation is approximately:




Q23. The energy lost per cycle is proportional to:




Q24. A lightly damped oscillator is one in which:




Q25. If damping is increased beyond the critical value, the system becomes:




Q26. In an underdamped system, the number of oscillations before coming to rest depends on:




Q27. Which of the following correctly expresses the total energy of a damped oscillator?




Q28. The damping coefficient has SI units of:




Q29. If the damping coefficient is doubled, the amplitude decay rate:




Q30. When the damping is removed completely, the motion becomes:




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