# Particle-Spring Systems

*Fear always springs from ignorance.*(Ralph Waldo Emerson, American Poet, Lecturer and Essayist, 1803-1882)

Let be the position at time of a unit point mass or *particle* moving without friction along a line subject to a *linear spring force* , see Intro to Springs.

Newton’s equations of motion take the form:

- .

# Simulator

Your first computational simulation:

- particle-spring harmonic oscillator
- output: position-velocity
- code: time stepping

# Watch

# To Think About

- How does a spring function?
- How to motivate that spring force is proportional to elongation?

# Conservation of Total Energy

The total energy a particle connected to a linear spring modeled by and is defined by

- .

Let us now prove that the *Trapezoidal Method* conserves the total energy. This follows by multiplying the time-stepping equations

by and , respectively, to get by summation and reordering (using that ),

which expresses conservation of the total energy as .

We understand that as the particle moves back and forth, kinetic energy is transformed into elastic energy stored as the spring stretches or compresses, which is transformed back into kinetic energy as the stretching and compression is eased.

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