Damped Harmonic Oscillator

Definition

A damped harmonic oscillator is a physical system that exhibits oscillatory motion, but with a decreasing amplitude over time due to the presence of a damping force. This type of oscillator is commonly encountered in various areas of physics, including mechanics, electronics, and acoustics.

5 Must Know Facts For Your Next Test

1. The motion of a damped harmonic oscillator is described by a second-order linear differential equation that includes a damping term.
2. The damping force in a damped harmonic oscillator can be caused by various mechanisms, such as friction, air resistance, or internal energy dissipation.
3. The amplitude of the oscillations in a damped harmonic oscillator decreases exponentially over time, with the rate of decay determined by the damping coefficient.
4. The energy of a damped harmonic oscillator is continuously dissipated due to the damping force, leading to a gradual loss of mechanical energy.
5. The behavior of a damped harmonic oscillator can be classified into three regimes: underdamped, critically damped, and overdamped, depending on the relative strength of the damping force.

Review Questions

• Explain how the damping force affects the motion of a harmonic oscillator.
  • The damping force in a harmonic oscillator causes the amplitude of the oscillations to decrease over time. This is because the damping force opposes the motion of the oscillator, dissipating its mechanical energy. The rate of decay of the oscillation amplitude is determined by the damping coefficient, with larger damping leading to a faster decrease in amplitude. The presence of damping also affects the frequency of the oscillations, which may be different from the natural frequency of the undamped system.
• Describe the different regimes of a damped harmonic oscillator and how they are characterized.
  • The behavior of a damped harmonic oscillator can be classified into three regimes based on the relative strength of the damping force: underdamped, critically damped, and overdamped. In the underdamped regime, the system oscillates with a decreasing amplitude, eventually coming to rest. In the critically damped regime, the system returns to equilibrium without oscillating, exhibiting a single, non-oscillatory response. In the overdamped regime, the system also returns to equilibrium without oscillating, but the response is slower and more gradual compared to the critically damped case. The classification of the damped oscillator depends on the value of the damping coefficient relative to the natural frequency of the undamped system.
• Explain how the conservation of energy principle applies to a damped harmonic oscillator, and describe the fate of the dissipated energy.
  • In a damped harmonic oscillator, the principle of conservation of energy is not strictly obeyed due to the presence of the damping force. The mechanical energy of the system, which is the sum of the kinetic and potential energies, is continuously dissipated by the damping force. This dissipated energy is converted into other forms, such as heat or sound, and is no longer available for the oscillatory motion. The rate of energy dissipation is proportional to the damping coefficient and the square of the velocity of the oscillator. As the oscillations decay, the mechanical energy of the system gradually decreases, and the dissipated energy is ultimately lost to the environment in the form of thermal energy or other forms of energy.

Related terms

Undamped Harmonic Oscillator: A harmonic oscillator without any damping force, where the oscillations continue indefinitely with a constant amplitude.

Critically Damped Oscillator: A damped harmonic oscillator where the damping force is just strong enough to prevent the system from oscillating, resulting in a single, non-oscillatory return to equilibrium.

Overdamped Oscillator: A damped harmonic oscillator where the damping force is so strong that the system does not oscillate, but instead, returns to equilibrium in a smooth, non-oscillatory manner.