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Unit 6

6.2 Energy of a Simple Harmonic Oscillator

2 min readβ€’june 8, 2020

Kanya Shah


Enduring Understanding 5.B

The energy of a system is conserved.

Essential Knowledge 5.B.2

A system with internal structure can have internal energy, and changes in a system’s internal structure can result in changes in internal energy

Essential Knowledge 5.B.3

A system with internal structure can have potential energy. Potential energy exists within a system if the objects within that system interact with conservative forces.

Essential Knowledge 5.B.3

The internal energy of a system includes the kinetic energy of the objects that make up the system and the potential energy of the configuration of objects that make up the system.

Energy in Simple Harmonic Oscillators

This topic is pretty much just an application of the energy types and conversions we covered in Unit 4: Energy. The main idea is that through SHM, the energy is converted from potential to kinetic and back again throughout the motion. The maximum potential energy occurs when the spring is stretched (or compressed) the most, and the maximum kinetic energy occurs at the equilibrium point.Β 
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Fo.PNG?alt=media&token=09fe7d2d-56c7-4a21-9e15-d7e12dc11a4e

Image Credit

Here’s an example using a mass on a spring, resting on a frictionless surface. In pictures A, C, and E, the energy is fully stored as potential energy in the spring. In pictures B and D, the mass is at the equilibrium position (x=0) and all the energy is now kinetic energy.
If we were to make a graph of energy vs time, it would look like this:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Fgr.PNG?alt=media&token=64eeb3f0-13d0-4243-8395-c9b3846942e1

Image Credit

A couple of things to notice in this graph above:
  1. The total energy is constant. This makes sense since there are no external forces to do work on the spring-mass system
  2. The potential energy and kinetic energy graphs are curves. Because of the squared term in the potential energy equation, we expect this. If the term is to the 1st power, then the graph would be linear.
  3. The potential energy is greatest when the position graph is at its maximum. The Kinetic Energy is greatest when the velocity graph is at its maximum.

Resources:

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