Oscillations

A pendulum is a basic example of an oscillating body.
Equilibrium position: Lowest position of the pendulum, a state of rest.
Forces acting on the pendulum:
- Force of Gravity (V||s)
- Tension in the String (T⟂s)
Tangential component (T⟂s) acts as the restoring force pulling the bob back to equilibrium.
At midpoint, restoring force vanishes; inertia causes overshooting.
Restoring force is maximal at extreme positions and continually changes direction, leading to oscillatory motion back and forth along an arc.
Periodic Motion: Repeating itself at equal time intervals.
Oscillation Definition: Periodic fluctuation in a physical quantity around the equilibrium value.
Types of quantities:
- Mechanical: Linear & angular displacement.
- Non-mechanical: Voltage, current, electric/magnetic fields, TV signals, X-ray/UV rays.
Harmonic Motion: Any motion repeating at regular intervals according to a sinusoidal law, such as motion of a mass under restoring force.
In harmonic motion, force is directly proportional to displacement.

Definition: Oscillatory motion where force is directly proportional to displacement.
Can involve various physical quantities like electrical currents and fields in circuits.
S.H.M. employs sinusoidal motion models (e.g., mass on a spring).
Force is characterized by Hooke's Law:
- F = -kx (where k is elastic constant, x is displacement).
- Negative sign indicates force is opposite to displacement direction.
Equation indicative of motion:
- Differential equation for simple harmonic motion:
  - m ··x + kx = 0
  - or
  - ··x + ω²x = 0, with ω = √(k/m) being the angular frequency.
Time Period (T): Time taken for one complete oscillation.
- T = 2π/ω = 2π√(m/k)
- Independent of amplitude; heavier masses drop period, while stiffer springs increase frequency.

Displacement:
- General solution: x = Asin(ωt + φ)
  - A is amplitude, varies periodically between -A and +A.
  - Phase angle is φ.
Velocity & Acceleration:
- Velocity: u = Aωcos(ωt + φ).
  - Periodic variation from +ωA to -ωA.
- Acceleration: a = -ω²x, varies between -ω²A and +ω²A.
Time Period:
- Determined by mass m and stiffness k, not by amplitude.

Total Energy: Constantly exchanged between potential & kinetic energy.
- Kinetic energy (Ek):
  - Ek = 1/2 mv² = 1/2 k(A² - x²)
- Potential energy (U):
  - U = 1/2 kx².
- Total Energy (E):
  - E = Ek + U = 1/2 kA² = constant.
Graphical Representation: P.E. is parabolic; K.E. is inversely parabolic.

Oscillations are damped through resistance (e.g., air, friction) resulting in decreased amplitude.
Damping: Energy loss through conversion to heat.
Damping Forces:
1. Restoring force proportional to displacement.
2. Frictional force proportional to velocity.
Equation of motion integrates damping:
- ma + mv' + kx = 0, rearranging yields damped harmonic motion equation.
Types of damping:
- Weak Damping: Oscillations are nearly sinusoidal, frequencies near natural frequency.
- Heavy Damping: System returns to equilibrium without oscillations.
- Critical Damping: System approaches equilibrium quickly without overshoot.

Defined as oscillations prompted by an external periodic force; differs from natural frequency.
Key forces include:
- Restoring force from elasticity.
- Damping force from resistance.
- Driving force from external periodic application (e.g., F = Fo sin(ωf t)).
Steady-state condition achieved after transient behavior:
- Steady-state solution:
  - x = A sin(ωf t - φ).
  - Amplitude A depends on the ratio of frequencies and damping.
Quality of System (Q): Measure of narrowness in frequency response; impacts resonance behavior.

Understanding of oscillations, particularly simple harmonic motion, damped conditions, and forced oscillations, essential for various applications in physics and engineering.