Unit 7 · Topic 07

Simple harmonic motion

SHM is recognized by the linear restoring acceleration a=−ω₀²x.

Unit 7Simple harmonic motionOpen navigation

Concept 01

Acceleration toward equilibrium

Essential The minimum you should retain

In SHM a=−ω₀²x: a and x have opposite signs except at equilibrium.

UnderstandInterpret and connect

At the extrema |a| is greatest and v=0.

DeepenFormulation and conditions

The constant ω₀² fixes the restoring proportionality.

ExploreConnections for further study

Linearity is common near smooth potential-energy minima.

Concept 02

Position, velocity, and acceleration

Essential The minimum you should retain

x=Acos(ω₀t+φ) x=A\cos(\omega_0t+\phi) , v=−Aω₀sin(ω₀t+φ), and a=−ω₀²x.

UnderstandInterpret and connect

v is a quarter cycle out of phase with x.

DeepenFormulation and conditions

Differentiation preserves signs and shows that a opposes x.

ExploreConnections for further study

Phase φ selects the initial condition without changing ω₀.

Worked example

Mass–spring oscillator

m=2.0 kg, k=200 N/m, and A=0.10 m.

Given
  • m=2.0 kg
  • k=200 N/m
  • A=0.10 m
Target

Find ω₀, T, and v_max.

  1. Frequency

    ω₀=sqrt(k/m)=10 rad/s.

  2. Period

    T=2π/ω₀≈0.628 s.

  3. Speed

    v_max=Aω₀.

  4. Result

    v_max=1.00 m/s.

Conclusion

ω₀=10 rad/s, T≈0.628 s, and v_max=1.00 m/s.

Mathematical relation

SHM state

x=Acos(ω₀t+φ),v=−Aω₀sin(ω₀t+φ),a=−ω₀2x x=A\cos(\omega_0t+\phi),\ v=-A\omega_0\sin(\omega_0t+\phi),\ a=-\omega_0^2x
Represents

Harmonic position, velocity, and acceleration.

Physical interpretation

Acceleration always opposes displacement.

DeepenVariables, conditions, and checks

Variables

A
nonnegative amplitude; usual unit: m
φ
initial phase; usual unit: rad

Conditions of application

  • Ideal simple harmonic motion.

Dimensional check

m, m/s, and m/s².

acceleration opposes displacement

Three normalized curves versus phase.
  • x/A
  • v/(Aω₀)
  • a/(Aω₀²)

x, v, and a have phase shifts and signs fixed by differentiation.

Concept 03

Natural frequency

Essential The minimum you should retain

For F=−kx, ω₀=sqrt(k/m) and T=2πsqrt(m/k).

UnderstandInterpret and connect

Increasing k raises ω₀; increasing m lowers it.

DeepenFormulation and conditions

In the ideal linear model ω₀ does not depend on A.

ExploreConnections for further study

Spring nonlinearities limit this independence.

Mathematical relation

Mass–spring oscillator

ω0=km,T=2πmk \omega_0=\sqrt{k/m},\quad T=2\pi\sqrt{m/k}
Represents

Natural frequency and period of an ideal spring.

Physical interpretation

ω₀ is independent of A in the linear model.

DeepenVariables, conditions, and checks

Variables

k
spring constant; usual unit: N/m
m
mass; usual unit: kg

Conditions of application

  • F=−kx; ideal linear spring.

Dimensional check

rad/s and s.

Three mass positions show F=−kx toward equilibrium.x<0mx>0FFx=0

On both sides of equilibrium, the restoring force points toward x=0.

Concept 04

When it works

Essential The minimum you should retain

SHM requires net force approximately proportional to −x.

UnderstandInterpret and connect

The origin must coincide with model equilibrium.

DeepenFormulation and conditions

Appreciable friction or external forces change the equation.

ExploreConnections for further study

Many systems admit SHM only for small displacements.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Giving a the same sign as x.

In SHM a=−ω₀²x.

Making ω₀ depend on A for an ideal linear spring.

ω₀=sqrt(k/m), independent of A in that model.