JEE Physics Notes: Oscillations

Introduction

Oscillations are repetitive back-and-forth motions around a central equilibrium position. They play a crucial role in various physical systems, including pendulums, springs, and even quantum mechanics.

1. Types of Oscillations

• Free Oscillations: Occur when a system oscillates without any external force after being displaced (e.g., simple pendulum).
• Damped Oscillations: Oscillations in which the amplitude decreases over time due to friction or resistance (e.g., pendulum in a viscous medium).
• Forced Oscillations: Occur when an external periodic force is applied to a system (e.g., a child pushing a swing).
• Resonance: When the frequency of an external force matches the natural frequency of the system, leading to maximum amplitude (e.g., bridges vibrating under wind force).

2. Simple Harmonic Motion (SHM)

• Definition: A special type of periodic motion in which restoring force is directly proportional to displacement and acts towards the mean position.
• Equation of SHM: F = -kx (Hooke’s Law).
• Differential Equation: m d²x/dt² + kx = 0.
• General Solution: x(t) = A sin(ωt + φ), where:
  • A = Amplitude
  • ω = Angular frequency (ω = 2π/T)
  • φ = Phase constant

3. Energy in SHM

• Kinetic Energy (KE): KE = (1/2) mω²(A² - x²).
• Potential Energy (PE): PE = (1/2) kx².
• Total Energy: E = (1/2) kA² (remains constant in SHM).

4. Time Period and Frequency

• Time Period (T): The time taken for one complete oscillation.
• Formula for Spring-Mass System: T = 2π√(m/k).
• Formula for Simple Pendulum: T = 2π√(l/g).
• Frequency (f): f = 1/T.

5. Damped Oscillations

• Definition: Oscillations in which the amplitude gradually decreases due to resistive forces like friction or air resistance.
• Equation: m d²x/dt² + b dx/dt + kx = 0, where b is the damping coefficient.
• Types:
  • Underdamped: Oscillations continue with decreasing amplitude.
  • Critically damped: Returns to equilibrium without oscillating.
  • Overdamped: Returns to equilibrium slowly without oscillating.

6. Forced Oscillations and Resonance

• Forced Oscillations: Occur when an external periodic force is applied to a system.
• Equation: m d²x/dt² + b dx/dt + kx = F₀ cos(ωt).
• Resonance: Maximum amplitude occurs when driving frequency equals natural frequency.

Conclusion

Oscillatory motion is fundamental in physics and engineering. Mastery of SHM, energy conservation, and resonance is essential for solving JEE Physics problems.