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7.1 Defining Simple Harmonic Motion (SHM)

Syllabus
2024
Topic
7.1
Level
—

Learning objectives

7.1A—Describe simple harmonic motionDescribe simple harmonic motion.• Simple harmonic motion is a special case of periodic motion.• SHM results when the magnitude of the restoring force exerted on an object is proportional to that object’s displacement from its equilibrium position. Derived equation: x- i. A restoring force is a force that is exerted in a direction opposite to the object’s displacement from an equilibrium position.- ii. An equilibrium position is a location at which the net force exerted on an object or system is zero.- iii. The motion of a pendulum with a small angular displacement can be modeled as simple harmonic motion because the restoring torque is proportional to the angular displacement. TOPIC 7.1 Defining Simple Harmonic Motion (SHM)

Recognize simple harmonic motion

Identify the defining relationship

Simple harmonic motion (SHM) is periodic motion produced by a restoring force whose magnitude is proportional to displacement from equilibrium and whose direction is opposite that displacement.

The equilibrium position is where the net force is zero. Measure signed displacement Δx\Delta x from this position; the force always points back toward it.

Map displacement to restoring direction

Position relative to equilibrium Δx\Delta x Restoring force FxF_x
Right/positive side Positive Negative, toward equilibrium
At equilibrium Zero Zero
Left/negative side Negative Positive, toward equilibrium

Use the linear restoring model

max=−kΔxma_x=-k\Delta x

Calculate force and acceleration

Example: a 0.50 kg0.50\,\text{kg} object is displaced +0.12 m+0.12\,\text{m} in a system with k=25 N m−1k=25\,\text{N}\,\text{m}^{-1}.

Fx=−kΔx=−(25)(0.12)=−3.0 NF_x=-k\Delta x=-(25)(0.12)=-3.0\,\text{N}

ax=Fx/m=−3.0/0.50=−6.0 m s−2a_x=F_x/m=-3.0/0.50=-6.0\,\text{m}\,\text{s}^{-2}.

Both negative signs mean the force and acceleration point back toward equilibrium.

Separate SHM from other periodic motion

Periodic motion alone is not enough: SHM requires the linear, opposite restoring relationship. A pendulum can be modeled as SHM only for small angular displacements, when restoring torque is proportional to angular displacement.