Periodic Motion Lecture

Definition of Periodic Motion

• Motion that repeats itself or oscillates back and forth.
• Examples: Mass-spring system, simple pendulum.

Mass-Spring System

• Components: Wall, spring, mass.
• Equilibrium Position: The natural length of the spring.
• Restoring Force: Opposes displacement from equilibrium, follows Hooke's Law: ( F_r = -kx ).
  • ( k ): Spring constant (N/m).
  • ( x ): Displacement from equilibrium.

Hooke's Law

• ( F = kx ): Force required to stretch/compress the spring.
• Negative Sign: Indicates restoring force direction opposite to displacement.

Simple Harmonic Motion (SHM)

• Characteristics:
  • Restoring force proportional to displacement.
  • Oscillations around equilibrium.
  • Velocity and acceleration vary sinusoidally.
• Energy:
  • Kinetic Energy (KE) is max at equilibrium, zero when fully stretched/compressed.
  • Potential Energy (PE) is max when fully stretched/compressed.

Calculations

• Force: ( F = kx )
• Work Done: ( W = \frac{1}{2} kx^2 )
• Mechanical Energy: Sum of kinetic and potential energy.

Dynamics of SHM

• Oscillator Behavior:
  • Max acceleration at extremes, max velocity at equilibrium.
• Frequency and Period:
  • Frequency ( f ): Number of cycles per second.
  • Period ( T ): Time per cycle, ( T = \frac{1}{f} ).

Damped Harmonic Motion

• Impact of Friction:
  • Reduces amplitude over time.
• Types:
  • Underdamped: Oscillates before stopping.
  • Overdamped: Returns to equilibrium slowly without oscillating.
  • Critically Damped: Returns quickly without oscillating.

Resonance

• Resonant Frequency: Natural frequency at which amplitude increases maximally when external force matches it.

Practice Problems

1. Calculating forces and displacement in springs.
2. Understanding energy transformations in oscillators.
3. Determining frequency and period in mass-spring systems.

Example Calculations

• Stretching/Compressing Force: ( F = k \times \text{displacement} ).
• Energy Stored: ( PE = \frac{1}{2} kx^2 ).
• Velocity & Acceleration in SHM: Derived using energy methods and Hooke's Law.

Important Equations

• Hooke's Law: ( F = -kx )
• Mechanical Energy: ( ME = KE + PE )
• Period: ( T = 2\pi \sqrt{\frac{m}{k}} )
• Frequency: ( f = \frac{1}{T} )

These notes summarize the key concepts of periodic motion, focusing on the mass-spring system, the mechanics of simple harmonic motion, and the effects of damping and resonance.