Lecture Notes on Periodic Motion and Harmonic Oscillations

Key Concepts

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

Mass-Spring System

• Equilibrium Position: The position where the spring is neither compressed nor stretched.
• Restoring Force: Force that brings the spring back towards its equilibrium position.
• Hooke's Law:
  • Formula: ( F_r = -kx )
  • ( k ) is the spring constant (N/m)
  • ( x ) is the displacement from the equilibrium.

Spring Constant

• Stiffness:
  • Higher ( k ) means stiffer spring.
  • ( k = \frac{F}{x} )

Practice Problems

1. Force to Stretch a Spring: Calculate the force needed to stretch a spring.

   • Example: 300 N/m spring stretched by 25 cm requires 75 N.

2. Distance with Given Force: Calculate how far a spring can be compressed or stretched given a force.

Simple Harmonic Motion (SHM)

• Characteristics:

  • Motion is sinusoidal (sine wave pattern).
  • Velocity and acceleration vary sinusoidally.

• Energy in SHM:

  • Potential Energy (PE): ( ,\frac{1}{2} kx^2 )
  • Kinetic Energy (KE): ( ,\frac{1}{2} mv^2 )

Mechanical Energy

• Total energy in a frictionless system remains constant.
• Mechanical Energy (ME) = KE + PE

Velocity and Acceleration in SHM

• Maximum Velocity & Accelerations:
  • ( v_{max} = \sqrt{\frac{k}{m}} A )
  • ( a_{max} = \frac{k A}{m} )

Frequency and Period

• Frequency (f): Number of oscillations per second (Hz).
• Period (T): Time to complete one cycle.
  • ( f = \frac{1}{T} )
  • For mass-spring system: ( T = 2\pi \sqrt{\frac{m}{k}} )

Energy Storage in Springs

• Elastic Potential Energy: Work required to stretch or compress a spring: ( U = \frac{1}{2} kx^2 )

Damped Harmonic Motion

• Damping: Reduction in amplitude over time due to friction or resistance.
• Types:
  • Over Damped: System returns to equilibrium slowly.
  • Critically Damped: System returns to equilibrium quickly.
  • Under Damped: System oscillates with decreasing amplitude.

Resonance

• Resonant Frequency: When driving frequency matches natural frequency, maximum amplitude is achieved.

Additional Problems

• Calculate using given mass, spring constants, and distances for various problems involving oscillations.

Equations of Motion

1. Position Function: ( x(t) = A \cos(2\pi f t) ) or ( x(t) = A \sin(2\pi f t) )
2. Velocity Function: ( v(t) = -v_{max} \sin(2\pi f t) )
3. Acceleration Function: ( a(t) = -a_{max} \cos(2\pi f t) )

Remember to practice solving problems involving these concepts and equations to strengthen understanding of harmonic motion and periodic systems.