484 |
Week 12: Gravity |
•Galileo208 (1564-1642 C.E.) is known as the Copernican heliocentric model’s most famous early defender, not so much because of the quality of his science as for his infamous prosecution by the Catholic church. In truth, Galileo was a contemporary of Kepler and his work was nowhere nearly as carefully done or mathematically convincing (or correct!) as Kepler’s, although using a telescope he made a number of important discoveries that added considerable further weight to the argument in favor of heliocentrism in general.
•Newton209 (1642-1727 C.E.) was the inheritor of the tremendous advances of Brahe, Descartes210 (1596-1650 C.E.), Kepler, and Galileo. Applying the analytic geometry invented by Descartes to the empirical laws discovered by Kepler and the kinematics invented by Galileo, he was able to deduce Newton’s Law of Gravitation:
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GM m |
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(1017) |
F = − |
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(a simplified form valid when mass M m, ~r are coordinates centered on the larger mass
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M , and F is the force acting on the smaller mass); we will learn a more precisely stated version of this law below. This law fully explained at the limit of observational resoution, and continues to mostly explain, Kepler’s Laws and the motions of the planets, moons, comets, and other visible astronomical objects! Indeed, it allows their orbits to be precisely computed and extrapolated into the distant past or future from a su cient knowledge of initial state.
•In Newton’s Law of Gravitation the constant G is a considered to be a constant of nature, and was measured by Cavendish211 in a famous experiment, thus (as we shall see) “weighing the planets”. The value of G we will use in this class is:
G = 6.67 × 10−11 |
N-m2 |
(1018) |
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You are responsible for knowing this number! Like g, it is enormously important and useful as a key to the relative strength of the forces of nature and explanation for why it takes an entire planet to produce a force on your body that is easily opposed by (for example) a thin nylon rope.
•The gravitational field is a simplification of Newton’s theory of gravitation that emerged over a considerable period of time with no clear author that attempts to resolve the problem Newton first addressed of action at a distance – the need for a cause for the gravitational force that propagates from one object to the other. Otherwise it is di cult to understand how one mass “knows” of the mass, direction and distance of its partner in the gravitational force! It is (currently) defined to be the gravitational force per unit mass or gravitational acceleration produced at and associated with every point in space by a single massive object. This field acts on any mass placed at that point and thereby exerts a force. Thus:
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GM m |
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• Important true facts about the gravitational field:
– The gravitational field produced by a (thin) spherically symmetric shell of mass M vanishes inside the shell.
208Wikipedia: http://www.wikipedia.org/wiki/Galileo.
209Wikipedia: http://www.wikipedia.org/wiki/Newton.
210Wikipedia: http://www.wikipedia.org/wiki/Descartes.
211Wikipedia: http://www.wikipedia.org/wiki/Cavendish.
Week 12: Gravity |
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– The gravitational field produced by this same shell equals the usual |
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~g(~r) = − |
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outside of the shell. As a consequence the field outside of any spherically symmetric distribution of mass is just
~g(~r) = − |
G M |
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(1022) |
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These two results can be proven by direct integration or by using Gauss’s Law for the gravitational field (using methodology developed next semester for the electrostatic field).
• The gravitational force is conservative. The gravitational potential energy of mass m in the field of mass M is:
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F~ · dℓ~ = − r |
(1023) |
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Um(~r) = −Z∞ |
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By convention, the zero of gravitational potential energy is at r0 = ∞ (in all directions).
•The gravitational potential is to the potenial energy as the gravitational field is to the force. That is:
V (~r) = m |
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~g · dℓ~ = − r |
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It as a scalar field that depends only on distance, it is the simplest of the ways to describe gravitation. Once the potential is known, one can always find the gravitational potential energy:
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Um(~r) = mV (~r) |
(1025) |
or the gravitational field: |
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~g(~r) = − V (~r) |
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or the gravitational force: |
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F m(~r) = −m V (~r) = m~g(~r) |
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•Escape velocity is the minimum velocity required to escape from the surface of a planet (or other astronomical body) and coast in free-fall all the way to infinity so that the object “arrives at infinity at rest”. Since U (∞) = 0 by definition, the escape energy for a particle is:
Eescape = K(∞) + U (∞) = 0 + 0 = 0 |
(1028) |
Since mechanical energy is conserved moving through the (presumed) vacuum of space, the total energy must be zero on the surface of the planet as well, or:
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mve2 − |
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ve = |
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2GM |
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(1030) |
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On the earth: |
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ve = r |
2GM |
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= p2gRe |
= 11.2 × 103 meters/second |
(1031) |
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(11.2 kilometers per second). This is also the most reasonable starting estimate for the speed with which falling astronomical objects, e.g. meteors or asteroids, will strike the earth. A large falling mass loses basically all of its kinetic energy on impact, so that even a fairly small asteroid can easily strike with an explosive power greater than that of a nuclear bomb, or many nuclear bombs. It is believed that just such a collision was responsible for at least the final Cretaceous extinction event that brought an end to the age of the dinosaurs some sixty million years ago, and similar collisions may have caused other great extinctions as well.
486 |
Week 12: Gravity |
• A (point-like) object in a plane orbit has a kinetic energy that can be written as:
K = Krot + Kr = |
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L2 |
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mvr2 |
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The total mechanical energy of this object is thus: |
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mvr2 |
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E = K + U = |
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2mr2 |
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is constant in this expression. |
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L for an orbit (in a central force, recall) is constant, hence L |
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The total energy and the angular momentum thus become convenient ways to parameterize the orbit.
•The e ective potential energy is of a mass m in an orbit with (magnitude of) angular momentum L is:
U ′(r) = |
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2mr2 |
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and the total energy can be written in terms of the radial kinetic energy only as: |
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E = |
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(1035) |
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This is a convenient form to use to make energy diagrams and determine the radial turning points of an orbit, and permits us to easily classify orbits not only as ellipses but as general conic sections. The term L2/2mr2 is called the angular momentum barrier because it’s negative derivative with respect to r can be interpreted as a strongly (radially) repulsive pseudoforce for small r.
•The orbit classifications (for a given nonzero L) are:
–Circular: Minimum energy, only one permitted value of rc in the energy diagram where E = U ′(rc).
–Elliptical: Negative energy, always have two turning points.
–Parabolic: Marginally unbound, E = 0, one radial turning point. This is the “escape orbit” described above.
–Hyperbolic: Unbound, E > 0, one radial turning point. This orbit has enough energy to reach infinity while still moving, if you like, although a better way to think of it is that its asymptotic radial kinetic energy is greater than zero.