Central Forces - physics.indiana.edudermisek/CM_13/CM-2-1p.pdf · Central Forces based on FW-3 ......
Transcript of Central Forces - physics.indiana.edudermisek/CM_13/CM-2-1p.pdf · Central Forces based on FW-3 ......
Central Forcesbased on FW-3
Motion of a particle with mass m subject to a central force:
)
conservative∇ x F = 0
no torque, r x F = 0 l = r x p = const.
It is convenient to work with polar coordinates:
(motion is planar, perpendicular to l)
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Conservation of energy:
Conservation of angular momentum:
first integral (involves only 1st time derivatives)
l = r x p = const.
(homework, problem 1.6)
motion in a small interval of time dt:
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Kepler’s second law:An imaginary line drawn from each planet to the Sun sweeps out equal areas in equal times.
(a simple consequence of the conservation of the angular
momentum for central forces)
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.
Effective potential:
Analyzing the motion:the angular momentum piece acts as
repulsive potential, centrifugal barrier
one dimensional potential:
E � Veff =1
2mr2 � 0
E � Veff
turning points
E = Veff (r0).
minimum energy, circular orbit
..
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Solution to the trajectory:
invert to find r(t)
plug in r(t) to find ϕ(t)
In practice this is often too complicated, but we can always find the geometric orbit r(ϕ) (at least numerically):
invert to find r(ϕ)
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Kepler’s problem (planetary motion):
Special case when the central force is gravitational force:
from the general solution:
we find:
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can be set to 0
straightforward manipulations:
Orbit solution:
For negative energy, orbits are bound and represent an ellipse.
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Ellipse:constant total distance from two foci
located at x=+f and x=-f:
eccentricity
equivalent definition:
in polar coordinates with the origin at the left focus:
follows from the law of cosines:
r
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Orbit solution: ellipse in polar coordinates with the origin at the left focus:
The orbit of a particle with negative energy in a gravitational field is an ellipse with the center at one focus.
Kepler’s first law:
energy of an orbit does not depend on angular momentum
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The square of an orbital period is proportional to the cube of a. Kepler’s third law:
Kepler’s 2nd law=
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