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

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

• The equilibrium position will often be labeled with $x=0$

• Displacement, $∆x$, is measured from the equilibrium position

• 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:

$m{a}_x=-k\mathrm{\Delta }x$

• 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:

$\overrightarrow{F_s}=-k\mathrm{\Delta }\overrightarrow{x}$

• Where:

  • ${\overrightarrow{F}}_s$ = spring's force measured in $\mathrm{N}$

  • $k$ = spring constant of the spring, measured in $\mathrm{N}/\mathrm{m}$

  • $\mathrm{\Delta }\overrightarrow{x}$ = displacement of the object from the equilibrium position, measured in $\mathrm{m}$

• 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:

${\overrightarrow{F}}_{net}=m\overrightarrow{a}$

• Where:

  • ${\overrightarrow{F}}_{net}$ = net force on the mass, measured in $\mathrm{N}$

  • $m$ = the object's mass, measured in $\mathrm{kg}$

  • $\overrightarrow{a}$ = the object's acceleration, measured in $\mathrm{m}/{\mathrm{s}}^2$

Step 2: Apply the specific conditions

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

$F_{net}=F_s$

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

$F_s=-k\mathrm{\Delta }x$

$F_{net}=m{a}_x$

• Where:

  • $a_x$ is the acceleration in the x direction, measured in $\mathrm{m}/{\mathrm{s}}^2$

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

$m{a}_x=-k\mathrm{\Delta }x$

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