Reading:Taylor 4.6 (Thornton and Marion 2.6) Giancoli 8.9 PH421: Oscillations – lecture 1 1.
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Transcript of Reading:Taylor 4.6 (Thornton and Marion 2.6) Giancoli 8.9 PH421: Oscillations – lecture 1 1.
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Reading: Taylor 4.6 (Thornton and Marion 2.6)
Giancoli 8.9
PH421: Oscillations – lecture 1
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Goals for the pendulum module:(1) CALCULATE the period of oscillation if we know the
potential energy (using exact and approximate forms); specific example is the pendulum
(2) MEASURE the period of oscillation as a function of oscillation amplitude
(3) COMPARE the measured period to models that make different assumptions about the potential
(4) PRESENT the data and a discussion of the models in a coherent form consistent with the norms in physics writing
(5) CALCULATE the (approximate) motion of a pendulum by solving Newton's F=ma equation
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How do you calculate how long it takes to get from one point to another?
Separation of x and t variables!
But what if v is not constant?
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Suppose total energy is CONSTANT (we have to know it, or be able to find out what it is)
The case of a conservative force
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Classical turning points
xL xR
B
x0
Example: U(x) = ½ kx2 , the harmonic oscillator
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xL xRx0
Symmetry - time to go there is the same as time to go back (no damping)
SHO - symmetry about x0
xL -> x0 same time as for x0 -> xR
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xL=-A xR=Ax0
SHO - do we get what we expect?
Another way to specify E is via the amplitude A
Independent of A!
x0=0
http://www.wellesley.edu/Physics/Yhu/Animations/sho.html
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xL=-A xR=Ax0
You have seen this before in intro PH, but you didn't derive it this way. 8
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Period of SHO is INDEPENDENT OF AMPLITUDEWhy is this surprising or interesting?
As A increases, the distance and velocity change. How does this affect the period for ANY potential?
A increases -> further to travel -> distance increases -> period increases
A increases -> more energy -> velocity increases -> period decreases
Which one wins, or is it a tie?
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Look at example of simple pendulum (point mass on massless string). This is still a 1-dimensional problem in the sense that the motion is specified by one variable,
Your lab example is a plane pendulum. You will have to generalize: what length does L represent in this case? 10
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Plot these to compare to SHO to pendulum…… 11
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Period increases because v(x) decreases (E decreases) for the same amplitude. Effect is magnified for larger amplitudes. 12
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Equivalent angular version?
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Integrate both sides
E and U() are known - put them in
Resulting integral• do approximately by hand using series expansion (pendulum period worksheet on web page)OR• do numerically with Mathematica (notebook on web page) 14
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xL xR
B
x0
“Everything is a SHO!”
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