Simple Harmonic Motion Examples

Simple harmonic motion is all around us, from swinging pendulums to vibrating strings. These examples illustrate how forces like gravity and elasticity create predictable, oscillating movements. Understanding these concepts is key in AP Physics 1, connecting theory to real-world applications.

1. Simple pendulum

   • Consists of a mass (bob) attached to a string or rod of fixed length.
   • The motion is periodic and can be described by the formula for the period: ( T = 2\pi\sqrt{\frac{L}{g}} ), where ( L ) is the length and ( g ) is the acceleration due to gravity.
   • The restoring force is gravity, which acts to bring the bob back to its equilibrium position.
   • The motion is simple harmonic for small angles (less than about 15 degrees).

2. Mass on a spring

   • A mass attached to a spring exhibits oscillatory motion when displaced from its equilibrium position.
   • The period of oscillation is given by ( T = 2\pi\sqrt{\frac{m}{k}} ), where ( m ) is the mass and ( k ) is the spring constant.
   • The restoring force is provided by Hooke's Law: ( F = -kx ), where ( x ) is the displacement from equilibrium.
   • Energy oscillates between potential energy in the spring and kinetic energy of the mass.

3. Torsional pendulum

   • A mass suspended from a wire or rod that twists about its axis when displaced.
   • The period of oscillation depends on the moment of inertia and the torsional constant of the wire: ( T = 2\pi\sqrt{\frac{I}{\kappa}} ).
   • The restoring torque is proportional to the angle of twist, similar to linear systems.
   • Commonly used in experiments to measure properties of materials.

4. Vibrating string or guitar string

   • A string fixed at both ends vibrates to produce sound, with nodes and antinodes forming along its length.
   • The fundamental frequency is determined by the length, tension, and mass per unit length of the string.
   • Harmonics can be produced, leading to a rich sound spectrum.
   • The wave equation describes the motion of the string: ( v = \sqrt{\frac{T}{\mu}} ), where ( T ) is tension and ( \mu ) is mass per unit length.

5. Oscillating liquid in a U-tube

   • A liquid in a U-tube oscillates when displaced from its equilibrium level.
   • The motion is driven by gravity and the restoring force is due to the difference in height of the liquid columns.
   • The period of oscillation can be approximated by ( T = 2\pi\sqrt{\frac{L}{g}} ), where ( L ) is the effective length of the liquid column.
   • Demonstrates principles of fluid dynamics and harmonic motion.

6. Vibrating tuning fork

   • A tuning fork vibrates at a specific frequency when struck, producing sound waves.
   • The frequency is determined by the material, shape, and dimensions of the fork.
   • The motion is a form of simple harmonic motion, with the prongs acting as oscillators.
   • Used as a standard for tuning musical instruments due to its consistent frequency.

7. Oscillating electrical circuit (LC circuit)

   • Consists of an inductor (L) and a capacitor (C) that store energy in magnetic and electric fields, respectively.
   • The circuit oscillates at a natural frequency given by ( f = \frac{1}{2\pi\sqrt{LC}} ).
   • Energy oscillates between the inductor and capacitor, similar to mechanical systems.
   • Damped oscillations can occur due to resistance in the circuit.

8. Vertical mass-spring system

   • A mass attached to a spring hangs vertically and oscillates when displaced.
   • The equilibrium position is affected by the weight of the mass, leading to a new equilibrium point.
   • The period of oscillation is still given by ( T = 2\pi\sqrt{\frac{m}{k}} ).
   • The system demonstrates both gravitational and elastic potential energy.

9. Horizontal mass-spring system

   • A mass on a frictionless surface attached to a spring oscillates horizontally when displaced.
   • The motion is purely horizontal, and the restoring force is provided by the spring.
   • The period of oscillation remains ( T = 2\pi\sqrt{\frac{m}{k}} ).
   • Ideal for studying simple harmonic motion without the influence of gravity.

10. Bungee jumper

    • A person attached to a bungee cord experiences oscillatory motion as they jump and rebound.
    • The motion involves both gravitational force and elastic force from the bungee cord.
    • The period of oscillation can be complex due to changing forces as the cord stretches and contracts.
    • Demonstrates real-world applications of simple harmonic motion principles.