phy191 exam
Jocelyn yang 
Ch.15_1.pptxOscillatory 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.
SHM Graphed
If the mass is pulled to x = +A and released from rest at t = 0, then A = the amplitude and Φ0 = 0
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frictionless surface
(spring equilibrium position)
We’ll handle the phase shift later, but it will still be useful to know where it is
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Whiteboard Problem 15-2
An air track glider is attached to a spring and oscillates between the 10cm mark and the 60cm mark on the track. The glider completes 10 oscillations in 33s.
Determine:
Period
Frequency
Angular Frequency
Amplitude
Maximum speed (LC)
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Energy
Like gravity, springs provide conservative forces
With no friction, all forces are conservative, so energy is conserved
Kinetic and potential energies change with time, but their sum is constant
Max potential?
Max kinetic?
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Energy Plots
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Whiteboard Problem 15-3
A 300g oscillator has a speed of 95.4cm/s when its displacement is 3.0cm, and 71.4cm/s when its displacement is 6.0cm.
What is the oscillator’s amplitude and maximum speed? (LC)
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You’re welcome to work as a table for this one
