Physics
Oscillations
Damped Harmonic Motion
11
⚡ Quick Summary
Damped harmonic motion occurs when a force opposing motion (damping force) is present along with the restoring force. This causes the amplitude of oscillations to decrease over time until the system comes to a halt.
x = A₀e^(-bt/2m)sin(ω't + δ)
ω' = √(ω₀² - (b/2m)²)
- In damped harmonic motion, a damping force acts opposite to the velocity, causing energy loss.
- If the damping is large, oscillations may cease entirely (critical damping). The system returns to equilibrium without overshooting.
- For small damping, the solution to the equation of motion is x = A₀e^(-bt/2m)sin(ω't + δ), where ω' = √(ω₀² - (b/2m)²).
- The amplitude decreases exponentially with time: A = A₀e^(-bt/2m).
Forced Oscillation and Resonance
11
⚡ Quick Summary
Forced oscillation occurs when an external periodic force is applied to an oscillating system. The system eventually oscillates at the frequency of the applied force. Resonance occurs when the applied frequency is close to the natural frequency of the system, leading to a large amplitude of oscillation.
F = F₀sin(ωt)
m(dv/dt) = -kx - bv + F₀sin(ωt)
x = A sin(ωt + φ)
A = (F₀/m) / √((ω₀² - ω²)² + (bω/m)²)
ω₀ = √(k/m)
ω' = √(ω₀² - (b/2m)²)
- Forced oscillation involves an applied periodic force F = F₀sin(ωt) in addition to restoring and damping forces.
- The equation of motion is m(dv/dt) = -kx - bv + F₀sin(ωt).
- After some time, the body oscillates with the frequency ω of the applied force: x = A sin(ωt + φ).
- The amplitude of forced oscillation is given by A = (F₀/m) / √((ω₀² - ω²)² + (bω/m)²).
- Resonance occurs when the applied frequency ω is close to the natural frequency ω₀, resulting in a maximum amplitude. For small damping, ω' ≅ ω₀.