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7.4 Energy of Simple Harmonic Oscillators

Syllabus
2024
Topic
7.4
Level
—

Learning objectives

7.4A—Describe the mechanical energy of a system exhibiting SHMDescribe the mechanical energy of a system exhibiting SHM.• The total energy of a system exhibiting SHM is the sum of the system’s kinetic and potential energies. Relevant equation: EUtotal =+K• Conservation of energy indicates that the total energy of a system exhibiting SHM is constant.• The kinetic energy of a system exhibiting SHM is at a maximum when the system’s potential energy is at a minimum.• The potential energy of a system exhibiting SHM is at a maximum when the system’s kinetic energy is at a minimum.- i. The minimum kinetic energy of a system exhibiting SHM is zero.- ii. Changing the amplitude of a system exhibiting SHM will change the maximum potential energy of the system and, therefore, the total energy of the system. Relevant equation for a spring–object system: Ek= 1 total A2 2 TOPIC 7.4 Energy of Simple Harmonic Oscillators

Track energy through an SHM cycle

Hold total energy constant

Etotal=U+K=constantE_{\mathrm{total}}=U+K=\text{constant}

Match energy to position

Position Speed Kinetic energy KK Potential energy UU
Equilibrium, x=0x=0 Maximum Maximum Minimum
Between equilibrium and a turning point Intermediate Between min and max Between min and max
Turning point, x=±Ax=\pm A Zero Minimum: 00 Maximum: EtotalE_{\mathrm{total}}

Use the spring energy model

Etotal=12kA2U(x)=12kx2K(x)=Etotal−U(x)\begin{gathered}E_{\mathrm{total}}=\frac12kA^2\\ U(x)=\frac12kx^2\\ K(x)=E_{\mathrm{total}}-U(x)\end{gathered}

Calculate the energy split

Spring example: k=80 N m−1k=80\,\text{N}\,\text{m}^{-1} and amplitude A=0.20 mA=0.20\,\text{m}. At x=0.12 mx=0.12\,\text{m}:

Etotal=12(80)(0.20)2=1.60 JE_{\mathrm{total}}=\tfrac12(80)(0.20)^2=1.60\,\text{J}

U=12(80)(0.12)2=0.576 JU=\tfrac12(80)(0.12)^2=0.576\,\text{J}

K=1.60−0.576=1.02 JK=1.60-0.576=1.02\,\text{J} to three significant figures. The two contributions still total 1.60 J1.60\,\text{J}.

Predict amplitude effects

Amplitude change Spring total-energy change
A→2AA\to2A Etotal→4EtotalE_{\mathrm{total}}\to4E_{\mathrm{total}}
A→A/2A\to A/2 Etotal→Etotal/4E_{\mathrm{total}}\to E_{\mathrm{total}}/4

Do not confuse one form with the total

At equilibrium, potential energy is minimum but the system's total energy is not zero: kinetic energy is maximum. At a turning point, kinetic energy is zero but total energy remains stored as potential energy.