Harmonic oscillator equations mastering physics

Oscillatory Motion

There and back again.

4/9/2108

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Motion

Things can move in lines

They can also move around a fixed point

Sometimes, really far away things can change motion

We’re going to take a look at the lines again, but now over and over

And over again

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A Spring Refresher

Hop on to Mastering Physics and start the PhET Lab: Masses & Springs

Work on this with your group, but make sure each person completes the assignment

You’ll have about 15 minutes to finish the PhET now, and whatever isn’t finished will be do at 6 today

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Simple Harmonic Motion

Hereafter commonly referred to as SHM

SHO will refer to a simple harmonic oscillator

Simple? 🗸 Motion? 🗸 Harmonic? …

References harmonic functions, like sines and cosines

Variety of different descriptors, but first; dynamics

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Dynamics

Our first model of SHM will be a spring:

If we pull the mass, the spring will pull back…

This is a linear restoring force: linear as it varies with x, restoring since it points toward an equilibrium position

Anytime you have a linear restoring force, you can have SHM

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frictionless surface

(spring equilibrium position)

So the spring always pushes or pulls the mass back to x = 0, the equilibrium position of the spring.

The Equation

Let us see what the forces acting on the mass are:

So we have an equation of motion, we just need to solve…

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Free Body Diagram for m:

(Nothing real interesting here.)

(What kind of animal is this?)

Second order linear differential equation

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What we have is a differential equation: some relationship between a function and some of the function’s derivatives

What we need to do is find a function that satisfies the relationship presented

We will do this by using the most sophisticated mathematical tool available: guessing

Solving The Equation

What we have is a differential equation: some relationship between a function and some of the function’s derivatives

What we need to do is find a function that satisfies the relationship presented

We will do this by using the most sophisticated mathematical tool available:

If you’ve guessed correctly, you’ve found the only solution

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Uniqueness theorem states that linear differential equations have only one unique solution.

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The Old College Try

We need a function whose second derivative is the negative of the function, with some other stuff

From calculus you should know that trigonometric functions have this property:

So, as a guess, why not a trig function?

Note that we aren’t necessarily dealing with angles, we’re using trig functions for the oscillating nature

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Whiteboard Problem 15-1

Show by direct substitution that our guess:

Is a solution of the differential equation:

For x(t) to be a solution, what must ω be equal to? (LC)

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Picking Things Apart

Consider our model:

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frictionless surface

(spring equilibrium position)

The solution:

Note: be careful using these equations in your calculator. The argument of the sine and cosine is in radians, and your calculator has to be set to radians.