Oscillations
1. Periodic and Oscillatory Motion
- Periodic Motion: Repeats at regular intervals (e.g. revolution of Earth).
- Oscillatory Motion: A type of periodic motion about an equilibrium position (e.g. pendulum).
- All oscillatory motions are periodic, but not all periodic motions are oscillatory.
2. Simple Harmonic Motion (SHM)
- A special type of oscillatory motion where restoring force is proportional to displacement and acts in the opposite direction.
- Restoring force: F = –kx
- Examples: Spring-mass system, simple pendulum (for small angles).
3. Characteristics of SHM
- Displacement (x): x = A sin(ωt + φ)
- Velocity (v): v = dx/dt = Aω cos(ωt + φ)
- Acceleration (a): a = –ω²x
- Time Period (T): T = 2π/ω
- Frequency (f): f = 1/T
- Amplitude (A): Maximum displacement from mean position.
4. Energy in SHM
- Kinetic Energy (K.E.): K.E. = (1/2)mv²
- Potential Energy (P.E.): P.E. = (1/2)kx²
- Total Energy: E = (1/2)kA² (Constant throughout SHM)
- Energy oscillates between kinetic and potential forms.
5. Spring-Mass System
- Time Period: T = 2π√(m/k)
- k = spring constant, m = mass
6. Simple Pendulum
- For small angles (θ < 10°), motion is SHM.
- Time Period: T = 2π√(l/g)
- l = length of pendulum, g = acceleration due to gravity
7. Damped Oscillations
- Amplitude of oscillation decreases over time due to energy loss (friction or air resistance).
- Motion dies out eventually.
8. Forced Oscillations
- Oscillations under the influence of an external periodic force.
- Steady-state amplitude depends on frequency of external force.
9. Resonance
- When frequency of external force matches natural frequency of system, amplitude becomes maximum.
- Useful in tuning radios and musical instruments.