Simple Harmonic Motion Notes

Simple Harmonic Motion

Simple Harmonic Motion (SHM)

  • SHM is a special case of periodic motion (repeating motion).
  • Defining characteristic: Restoring force.
  • SHM involves a force that brings the system back to equilibrium.
  • Two common cases:
    • Mass-spring oscillator
    • Simple pendulum

Mass-Spring Oscillators

  • Equilibrium: Spring is neither stretched nor compressed.
  • Stretching: Spring exerts a force to pull the object back toward equilibrium.
  • Compression: Spring exerts a force to push the object toward equilibrium.
  • The spring force is the restoring force because it always brings the system back to equilibrium.
  • Hooke’s Law: Fs=−kΔxF_s = -k\Delta x
    • FsF_s: Spring force
    • kk: Spring constant
    • Δx\Delta x: Displacement from equilibrium
    • Note: The force direction is always opposite to the displacement.

Simple Pendulum

  • Equilibrium: Pendulum bob is hanging straight downward.
  • Displacement: A restoring force pulls it back toward equilibrium.
  • Restoring force: A component of gravity.

Describing SHM - Period and Frequency

  • Frequency (ff):
    • Number of cycles per second (Hertz).
  • Period (TT):
    • Number of seconds per cycle (seconds).
  • Frequency and period are reciprocals of each other. f=1Tf = \frac{1}{T}

Period of a Pendulum

  • The equation for the period of a simple pendulum is: Tp=2πlgT_p = 2\pi \sqrt{\frac{l}{g}}

    • TpT_p: Period of the pendulum
    • ll: Length of the pendulum
    • gg: Acceleration due to gravity

  • As length of the pendulum increases, period increases.

  • As mass of the pendulum increases, period remains the same.

  • As acceleration due to gravity increases, period decreases.

Period of a Spring

  • The equation for the period of a spring: Ts=2πmkT_s = 2\pi \sqrt{\frac{m}{k}}
    • TsT_s: Period of the spring
    • mm: Mass
    • kk: Spring constant
  • As mass increases, period increases.
  • As the spring constant increases, period decreases.
  • As the acceleration due to gravity increases, period remains the same.

Analyzing the Motion of a Simple Harmonic Oscillator

  • Analyzing points along the cycle of the mass/spring oscillator:
    • Maximum compression (displacement = -A)
    • Equilibrium (displacement = 0)
    • Maximum extension (displacement = +A)

Maximum Compression

  • Displacement = -A
  • Force = −kx=−k(−A)=kA-kx = -k(-A) = kA (positive)
  • Acceleration = positive
  • Velocity = 0
  • Mass is about to change direction.

Equilibrium

  • Displacement = 0
  • Force = −kx=−k∗0=0-kx = -k * 0 = 0
  • Acceleration = 0
  • Velocity = maximum (in the positive direction).

Maximum Extension

  • Displacement = +A
  • Force = −kx=−k∗+A=−kA-kx = -k * +A = -kA (negative)
  • Acceleration = negative
  • Velocity = 0
  • Mass is about to change directions.

Graphing SHM: x=Acos⁡(2πft)x = A \cos(2 \pi ft)

  • AA = Amplitude
    • Increasing A makes the graph taller.
    • Decreasing A makes the graph shorter.
  • ff = frequency
    • Increasing f makes the graph narrower.
    • Decreasing f makes the graph wider.

Analyzing an SHM Position vs Time Graph

  • At maximum displacement, Velocity is zero (Slope = zero).
  • At equilibrium Velocity is at a maximum (Slope is steepest).