Defining Simple Harmonic Motion (College Board AP® Physics 1: Algebra-Based)

Defining simple harmonic motion

What is simple harmonic motion?

• Simple harmonic motion (SHM) is a special case of periodic motion

• In SHM, an object oscillates regularly around an equilibrium position:

An equilibrium position is a location at which the net force exerted on an object is zero

• This is where displacement is defined as zero so the equilibrium position will often be labeled with

• Examples of systems that exhibit SHM:

  • An object attached to a spring oscillating horizontally across a frictionless surface (often called an object-ideal spring system)

  • A pendulum (object suspended from a string) with a small angular displacement

Systems in simple harmonic motion

Conditions for simple harmonic motion

• A restoring force is required for SHM:

A restoring force always acts in a direction opposite to the object’s displacement from an equilibrium position

• The condition for SHM is that the restoring force is proportional to the displacement of the object from the equilibrium position but in the opposite direction

Conditions for a pendulum in SHM

• A linear system, such as an object-ideal spring system, requires a restoring force in the opposite direction to linear displacement

• A pendulum, however, is better described as moving through angular displacement

• This means that, for a pendulum to undergo SHM, a restoring torque must be proportional to angular displacement but act in the opposite direction

  • These quantities are only proportional for small angular displacements

  • A pendulum can only be modeled as SHM when the angle is small

SHM in a pendulum

Derived equation

• Simple harmonic motion of an object-ideal spring system can be characterized by the equation:

• This shows that the acceleration of the object is proportional to the displacement and acts in the opposite direction

Object-ideal spring system

Derivation

Step 1: Identify the fundamental equations

• Recall Hooke's law:

• Where:

  • = spring's force measured in

  • = spring constant of the spring, measured in

  • = displacement of the object from the equilibrium position, measured in

• This describes the restoring force, as it acts in the opposite direction to the displacement vector

• Recall Newton's second law, which relates the force on an object with its mass and acceleration:

• Where:

  • = net force on the mass, measured in

  • = the object's mass, measured in

  • = the object's acceleration, measured in

Step 2: Apply the specific conditions

• The spring's restoring force is the only force that acts, therefore:

• Additionally, all forces act in the x direction so Hooke's law and Newton's second law can be rewritten as:

• Where:

  • is the acceleration in the x direction, measured in

• Substituting Hooke's law and Newton's second law into the above gives:

• This is the derived equation and does not appear on the equation sheet