Lecture on Periodic Motion and Simple Harmonic Motion

Introduction to Periodic Motion

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

Mass-Spring System

• Setup: Mass attached to a spring connected to a wall.
• Equilibrium Position: Initial position of the spring when no force is applied.
• Restoring Force: Force that pulls the spring back to its equilibrium position when stretched or compressed.
• Hooke's Law: ( F_r = -kx )
  • ( x ): Displacement from equilibrium.
  • ( k ): Spring constant (N/m).

Understanding Spring Constant (k)

• Units: Newtons per meter (N/m).
• Stiffness: Higher ( k ) means the spring is stiffer and harder to compress/stretch.

Practice Problems with Spring Force

• Example Calculation: Stretching a spring with given spring constant and displacement.
• Unit Conversion: Converting centimeters to meters for accurate calculations.

Energy in Mass-Spring Systems

• Potential Energy: Stored as ( U = \frac{1}{2} kx^2 ).
• Kinetic Energy (KE) and Potential Energy Relationship:
  • Fully stretched/compressed: Maximum potential energy, zero kinetic energy.
  • At equilibrium: Maximum kinetic energy, zero potential energy.
• Mechanical Energy: Total energy, constant if no friction.

Equations for Maximum Velocity and Acceleration

• Maximum Velocity: ( v_{max} = A \sqrt{\frac{k}{m}} ).
• Maximum Acceleration: ( a_{max} = \frac{kA}{m} ).

Oscillation Characteristics

• Amplitude (A): Maximum displacement from the equilibrium.
• Frequency (f) and Period (T):
  • Frequency: Number of oscillations per second (Hz).
  • Period: Time for one complete cycle.
  • Relationship: ( f = \frac{1}{T} ).

Impact of Changing Parameters

• Doubling Amplitude: Increases mechanical energy by a factor of four.
• Mass and Spring Constant Effect:
  • Increasing mass increases period.
  • Increasing spring constant decreases period.

Damped Harmonic Motion

• Damping: Reduction of amplitude due to friction.
• Types:
  • Under Damped: Oscillates with gradually decreasing amplitude.
  • Over Damped: Returns to equilibrium without oscillating.
  • Critically Damped: Quickly returns to equilibrium without oscillation.

Forced Vibrations and Resonance

• Resonance: Occurs when the force applied matches the natural frequency.
• Result: Maximum amplitude increase.
• Example: Pushing a swing at the right moment to increase swing height.

Graphical Representation and Equations in SHM

• Position Function: ( x(t) = A \cos(2\pi f t) ) or ( x(t) = A \sin(2\pi f t) ).
• Velocity and Acceleration Functions:
  • Velocity: Derived from position function.
  • Acceleration: Derived from velocity function.

Problem Solving in SHM

• Calculation of frequency, period, and maximum energy in oscillating systems.
• Understanding and applying conservation of energy: Equating mechanical energy to kinetic and potential energy.
• Use of trigonometric functions to describe motion: Different equations for different starting conditions.