Periodic Motion and Simple Harmonic Motion

Introduction

• Periodic Motion: Motion that repeats itself or oscillates back and forth.
• Simple Harmonic Motion (SHM): A type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.
• Examples: Mass-spring system and simple pendulum.

Mass-Spring System

• A mass attached to a spring oscillates back and forth around an equilibrium position.
• Restoring Force: Acts to return the system to equilibrium, given by Hooke's Law: ( F_r = -kx ).
  • k: Spring constant (N/m).
  • x: Displacement from equilibrium.
• Equilibrium Position: Point where net force is zero; at this point, acceleration is zero and velocity is at a maximum.
• Stiffness: A higher spring constant means a stiffer spring, requiring more force to stretch or compress.

Energy in SHM

• Kinetic Energy (KE): Maximum at equilibrium position; ( KE = \frac{1}{2}mv^2 ).
• Potential Energy (PE): Maximum when fully stretched or compressed; ( PE = \frac{1}{2}kx^2 ).
• Mechanical Energy (ME): Sum of KE and PE, constant if no friction present.

Calculations

• Force Required: ( F = kx ).
• Work Done: ( W = \int F dx ) or ( W = \frac{1}{2} k x^2 ).
• Max Velocity: ( v_{max} = \sqrt{\frac{k}{m}} \times A ).
• Max Acceleration: ( a_{max} = \frac{kA}{m} ).

Velocity and Acceleration as Functions of Time

• Position Function: ( x(t) = A \cos(2\pi ft) ).
• Velocity Function: ( v(t) = -V_{max} \sin(2\pi ft) ).
• Acceleration Function: ( a(t) = -A_{max} \cos(2\pi ft) ).

Period and Frequency

• Period (T): Time for one full cycle; ( T = \frac{2\pi}{\omega} ).
• Frequency (f): Number of cycles per second; ( f = \frac{1}{T} ).
• Relationship: ( T = 2\pi \sqrt{\frac{m}{k}} ), ( f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} ).

Effect of Changes in System Parameters

• Doubling Amplitude (A): Increases mechanical energy by a factor of 4, velocity, and acceleration by a factor of 2.
• Increasing Mass (m): Increases period (T) and decreases frequency (f).
• Increasing Spring Constant (k): Decreases period (T) and increases frequency (f).

Practice Problems

• Calculating the force required to stretch/compress a spring, work done, maximum kinetic and potential energy.
• Finding the frequency and period of oscillations for mass-spring systems.

Damping

• Damped Harmonic Motion: Occurs when friction is present, reducing amplitude over time.
• Types of Damping:
  • Underdamped: Oscillations decrease gradually.
  • Overdamped: Returns to equilibrium without oscillating.
  • Critically Damped: Returns to equilibrium as quickly as possible without oscillating.

Resonance

• Resonant Frequency: The frequency at which the amplitude of oscillation is maximized.
• Application: Pushing a swing at the right time (resonant frequency) increases oscillation amplitude efficiently.