race to the bottom brachistochrones and simple harmonic motion 1 race to the bottom
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Race to the BottomBrachistochrones and Simple Harmonic Motion
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Least Time Path from A to B
2x
y
A
B
g
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The Brachistochrone Problem
•Greek for “shortest time”•first expressed by Galileo•solved by multiple Bernoulli’s• led to calculus of variations and the Euler-Lagrange equations
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Formulation
If we choose a path, , we can compute the time it takes to follow the path from
and
where is called “arc-length”.
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Formulation (cont.)
The calculus of variations allows us to compute the shape of the path from A to B that makes
as small as possible.
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Formulation (cont.)
The answer is a cycloid:
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Some Example Times
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Shape Timestraight line 1.189quadratic 1.046cubic 1.019ellipse 1.007cycloid 1.003
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Isochronous
8x
y
A
B
gC
Time from C to B is the same as the time from A to B
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Simple Harmonic Motion
Suppose a particle is in a system with potential energy that has a minimum at .
9x=0
𝑉 (𝑥)
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Disturb the Particle
Represent as a power series
The force exerted by on the particle is
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Disturb the Particle (cont.)
At a minimum, . Further for x sufficiently close to zero (i.e., a small enough disturbance), we can accurately approximate F by
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Equation of Motion
So the equation of motion (Newton-II) for the particle is just
It’s general solution is
where .12
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Why Do We Obsess About SHM?
• The preceding line of argument applies to almost any system which is in equilibrium and is slightly disturbed.• An extremely large class of systems:• pendulums• the Earth after an earthquake• you and me• stars
• SHM solutions show up everywhere13
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When Good Methods Fail
The assumptions of SHM break down when the disturbance becomes too large.
Consider a simple pendulum:
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Large-amplitude Pendulum
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Huygens Pendulum
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period independent of amplitude
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Pendulum in Phase Space
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