phy191 exam
Jocelyn yang 
Ch.15_2.pptxPhase, Pendulums, & Damping
An investigation into swingers
4/11/2108
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Last Time…
Our model:
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frictionless surface
(spring equilibrium position)
The Phase Constant
If the block is pulled to A and released from rest, the phase constant is zero.
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frictionless surface
(spring equilibrium position)
A
phase
phase constant
The Phase Constant
What if the block isn’t released from rest at A?
There is still an oscillation between ±A, but now we need a phase constant
The phase constant exists to provide that shift away from A when t=0
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Note: This thing moves when you run the power point, the x location at t = 0 is no longer A
Example Problem (15-4)
The position vs. time graph for a SHO is shown below. What is the phase constant ?
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What does your
calculator say?
Which one is it? We
need a way to figure
this out.
T
T = 4 s
Finding Φ
To determine the sign of the phase constant, we’ll turn to uniform circular motion
Now what if the object doesn’t start on the x-axis?
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The x-component of the object’s position vector is:
Now, the x-component of the object’s position vector is:
Great, but what do I need to memorize here?
Nothing… but…
We know initial conditions x(0) and v(0)
So we can determine the correct quadrant for Φ
Finding Φ
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Consider that you can draw a circle and determine these things without memorizing them. The motion starts at the x axis, and the x velocity starts out negative and becomes positive once you start to come back around the circle. Furthermore, you can tell when the x velocity is increasing or decreasing in magnitude.
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Example Problem (15-4)
The position vs. time graph for a SHO is shown below. What is the phase constant ?
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T
For this problem:
slope < 0
Whiteboard Problem 15-5
Below is a velocity vs. time graph of a particle in simple harmonic motion.
What is the amplitude of the oscillation?
What is the phase constant? (LC)
What is the position at t = 0?
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Note this is a plot of velocity, it’s a little different than the one that we just did.
Tick… Tock…
Swing on over to Mastering Physics and start the PhET Lab: Pendulums
Work as a group, but submit individually.
Sometime before you finish the PhET, calculate the length of the pendulum in the lobby of Kreger (LC) using the equation below:
You’ll have about 20 minutes to finish the PhET now, and whatever isn’t finished will be due at 6 today
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A simple pendulum is a point mass on a massless string that can swing around a pivot point
The motion of the pendulum can be described with torque
The Simple Pendulum
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+
This is not SHM, but if we consider only small angles:
Whiteboard Problem 15-6
What is the period of a 1.0m long pendulum on:
Earth?
Venus? (LC)
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Pivot
point
Physical Pendula
A physical pendulum is a real object that can rotate about some pivot point:
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Center of
mass
The text uses the same steps that we used for a simple pendulum to show that for small angles, this is also SHM where:
Whiteboard Problem 15-7
A uniform rod of mass M and length L swings as a pendulum on a pivot at a distance of L/4 from one end of the rod. Find an expression for the frequency of the oscillation for small angles. (LC)
Your expression should contain only g, L, and numbers
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Chaos
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Double pendula exhibit chaotic behavior; the progression of the motion is very sensitive to initial conditions.
Other neat things: https://www.youtube.com/watch?v=B6vr1x6KDaY
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Damped Oscillations
Our model so far has excluded all non-conservative forces
So what’s the rub?
Here’s one model of a damped oscillator:
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FBD:
b = damping constant
This is not an easy differential equation to guess a solution for… we’ll just jump to the solution.
The Answer
The solution for this model of a damped oscillator is a decaying exponential:
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oscillating part
decaying part
The angular frequency
of the undamped
oscillator.
Whiteboard Problem 15-8
A 250g air track glider is attached to a spring with spring constant 4.0N/m. The damping constant due to air resistance, b, is 0.015kg/s. The glider is pulled out 20cm from equilibrium and released.
How many oscillations will it make during the time in which the amplitude decays to e-1 of its initial value? (LC)
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Driven Oscillations & Resonance
Nonconservative forces can take energy out of an oscillator, but they can also put energy in
The driving force should be applied with the same frequency as the oscillator; the oscillator’s natural frequency
When this condition is met, it is called resonance
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Examples include swinging, shattering glass, and the Tacoma narrows bridge
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Lavf54.63.104
Lavf54.63.104
Lavf54.63.104
