Simple Harmonic Motion Notes

Simple Harmonic Motion (SHM)

• SHM is a special case of periodic motion with a restoring force that brings the system back to equilibrium.

• Common examples: mass-spring oscillator and simple pendulum.

Mass-Spring Oscillators

• Equilibrium: Spring is neither stretched nor compressed.

• Stretching: Spring exerts a force to pull the object toward equilibrium.

• Compression: Spring exerts a force to push the object toward equilibrium.

• Hooke’s Law: Fs = -kΔx (force is always opposite displacement).

  • Note:  The direction of the force is always opposite the displacement 

Simple Pendulum

• Equilibrium: Pendulum bob hangs straight downward.

• Restoring force: Component of gravity pulling it back toward equilibrium.

Describing SHM

• Frequency (f): Number of cycles per second (Hertz).

• Period (T): Number of seconds per cycle.

• Relationship: Frequency and period are reciprocals of each other.

Period of a Pendulum

• Formula: T = 2π \sqrt{\frac{l}{g}}

• As length (l) increases, period increases.

• As mass increases, period remains the same.

• As gravity (g) increases, period decreases.

Period of a Spring

• Formula: T = 2π \sqrt{\frac{m}{k}}

• As mass (m) increases, period increases.

• As spring constant (k) increases, period decreases.

• As gravity increases, period remains the same.

Analyzing SHM

• Maximum Compression (Displacement = -A):

  • Force = +kA

  • Acceleration = positive

  • Velocity = 0

• Equilibrium (Displacement = 0):

  • Force = 0

  • Acceleration = 0

  • Velocity = maximum

• Maximum Extension (Displacement = +A):

  • Force = -kA

  • Acceleration = negative

  • Velocity = 0

Graphing SHM: x = Acos(2πft)

• A = Amplitude: Affects graph height.

• f = Frequency: Affects graph width.

Analyzing Position vs Time Graph

• At maximum displacement, velocity is zero (slope = zero).

• At equilibrium, velocity is at a maximum (slope is steepest).