Overview of equations and skills for the energy of simple harmonic oscillators, including how to find the elastic potential energy and kinetic energy over time. Understand how total energy, kinetic energy, and potential energy are all related.

Equations

EquationSymbol breakdownMeaning in words
Us=12kx2U_s = \dfrac{1}{2} k x ^2UsU_s is the elastic potential energy, kk is spring constant, and xx is length of extension or compression relative to the un-stretched length.The elastic potential energy is directly proportional to the square of the change in length and the spring constant.
ΔUg=mgΔy\Delta U_g = mg \Delta yΔUg\Delta U_g is change in gravitational potential energy, mm is mass, gg is the gravitational field strength, and Δy\Delta y is change in height.The change in gravitational potential energy is directly proportional to mass, gravitational field strength, and change in height.
K=12mv2K = \dfrac{1}{2} mv^2KK is translational kinetic energy, mm is mass, and vv is the speed.Translational kinetic energy is directly proportional to mass and the square of the speed.

How to find energy over time for a simple harmonic oscillator

Elastic potential energy

Elastic potential energy depends upon the position of our system, so a position vs. time graph can be used to find the elastic potential energy UsU_s over time for a simple harmonic oscillator. There are a few important points to note when comparing the position and energy graphs:
  • Us, maxU_\text {s, max} occurs when the system is at the maximum displacement of AA and A-A.
  • Us=0U_s=0 occurs when the system is at x=0x=0.

Kinetic energy

Kinetic energy KK depends upon the speed of a system, so a velocity vs. time graph can be used to find the kinetic energy over time for simple harmonic oscillator. There are a few important points to notice when comparing the velocity and energy graphs:
  • KmaxK_\text {max} occurs when the system is at its maximum speeds vmax|v_\text {max}| and vmax|-v_\text {max}|.
  • K=0K=0 occurs when v=0v=0.
Figure 2. A comparison of the velocity vs. time graph and kinetic energy vs. time graph for a simple harmonic oscillator.

Total energy

The total energy is the sum of the kinetic and elastic potential energy of a simple harmonic oscillator:
E=K+UsE=K+U_s
The total energy of the oscillator is constant in the absence of friction. When one type of energy decreases, the other increases to maintain the same total energy.
Figure 3. A graph of energy vs. time for a simple harmonic oscillator. This graph shows total energy EtotE_\text {tot} (purple), kinetic energy KK (red), and elastic potential energy UsU_s (blue).
There are a few important points to keep in mind about energy:
  • Us, maxU_\text {s, max} occurs when K=0K=0. This happens at the endpoints of the oscillation where the system momentarily stops (v=0v=0) at the maximum displacement.
  • KmaxK_\text {max} occurs at Us=0U_s=0. This is when the system is moving through the equilibrium position (x=0x=0) and has its maximum speed.
  • EtotE_\text {tot} is constant, so Etot=Kmax=Us, maxE_\text {tot} = K_\text {max} = U_\text {s, max}.

Learn more

To check your understanding and work toward mastering these concepts, check out our exercises:
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