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simulations.pendulum.title

simulations.pendulum.description

simulations.pendulum.parameters.title

simulations.pendulum.parameters.shortsimulations.pendulum.parameters.long
simulations.pendulum.parameters.weaksimulations.pendulum.parameters.strong
simulations.pendulum.parameters.highDampingsimulations.pendulum.parameters.noDamping

simulations.pendulum.explanation.title

simulations.pendulum.explanation.basicPhysics

simulations.pendulum.theory.title

simulations.pendulum.theory.introduction

simulations.pendulum.theory.energyConservation.title

simulations.pendulum.theory.energyConservation.description

  • simulations.pendulum.theory.energyConservation.potentialEnergy: U = mgh = mgL(1-cosθ)
  • simulations.pendulum.theory.energyConservation.kineticEnergy: K = ½mv² = ½mL²ω²
  • simulations.pendulum.theory.energyConservation.lowestPoint: simulations.pendulum.theory.energyConservation.lowestPointExplanation
  • simulations.pendulum.theory.energyConservation.highestPoint: simulations.pendulum.theory.energyConservation.highestPointExplanation

simulations.pendulum.theory.colorExplanation

simulations.pendulum.theory.periodRelation

simulations.pendulum.formulaDerivation.title

simulations.pendulum.formulaDerivation.periodFormula.title

simulations.pendulum.formulaDerivation.periodFormula.step1

Ft = -mg sin θ

simulations.pendulum.formulaDerivation.periodFormula.step2

mat = -mg sin θ

at = -g sin θ

simulations.pendulum.formulaDerivation.periodFormula.step3

at = L·(d²θ/dt²)

L·(d²θ/dt²) = -g sin θ

d²θ/dt² = -(g/L) sin θ

simulations.pendulum.formulaDerivation.periodFormula.step4

sin θ ≈ θ (当θ很小时)

d²θ/dt² ≈ -(g/L)·θ

simulations.pendulum.formulaDerivation.periodFormula.step5

θ = A·cos(ωt)

ω = √(g/L)

T = 2π/ω = 2π·√(L/g)

simulations.pendulum.formulaDerivation.periodFormula.conclusion

simulations.pendulum.formulaDerivation.smallVsLargeAngle.title

simulations.pendulum.formulaDerivation.smallVsLargeAngle.explanation1

T = 2π·√(L/g)·(1 + (1/16)·θ0² + ...)

simulations.pendulum.formulaDerivation.smallVsLargeAngle.explanation2

simulations.pendulum.formulaDerivation.smallVsLargeAngle.table.anglesimulations.pendulum.formulaDerivation.smallVsLargeAngle.table.errorsimulations.pendulum.formulaDerivation.smallVsLargeAngle.table.actualPeriod
5°< 0.1%≈ T0
10°≈ 0.5%≈ 1.005·T0
30°≈ 4.3%≈ 1.043·T0
90°≈ 18%≈ 1.18·T0

simulations.pendulum.formulaDerivation.smallVsLargeAngle.explanation3

simulations.pendulum.formulaDerivation.applications.title

simulations.pendulum.formulaDerivation.applications.explanation1

simulations.pendulum.formulaDerivation.applications.clock

simulations.pendulum.formulaDerivation.applications.clockExplanation

simulations.pendulum.formulaDerivation.applications.gravity

simulations.pendulum.formulaDerivation.applications.gravityExplanation

simulations.pendulum.formulaDerivation.applications.explanation2