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CAPSTONE Lecture 2
Gravity and its uses
07.06.2010
07/06/2010
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Lecture 2: Gravity and its uses
I. The laws of gravity
II. Galileo, g
III. Newton, G
IV. Orbital velocity
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Newton’s Laws of Gravity
From studies of the motion of bodies, Newton
concluded that three laws governed motion.
a) Every action on a body has an equal
and opposite reaction.
b) Bodies at rest stay at rest until acted
on by a force. Bodies in uniform motion
maintain that motion until acted on by a
force.
c) The force is the mass of a body times
its acceleration (uniform change in
velocity per unit time.
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--For gravity as a force, Newton concluded that F=GMm/r2, where G is a constant and M and m are the
inertial masses of two bodies separated by a
distance, r.
--Newton convinced himself that the ability of a body
to attract another body was directly proportional to the
intertial mass of the same body and that the
proportionality constant was G.
--He also showed that for an object far from the
attracting body, the latter could be treated as a point
source of gravity and that neither the shape, density
or type of material affected this law.
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Circular velocity for the
gravitational force
Huygens showed (mid 17th century) that for
objects in circular orbits, no matter the nature of
the force constraining one body to orbit another,
F=-mv2/r,
where v is the velocity of the orbiting body and
the other terms are as defined above. The
minus sign is a convention that means an
attractive force (which will act to reduce the
distance between the two bodies.)
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Equating Huygen’s expression with
Newton’s expression for the specific force of
gravity:
-GMm/r2=-mv2/r, where M and m are the two
masses involved, G is the universal
gravitational constant, r is the separation of
the two bodies.
So, the orbital velocity for perfect circular
motion is v=(GM/r)1/2 , assuming m<<M.
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BIG G and little g
Newton’s force law, F=-GmM/r2, reduces to F=mg, where g=GM/r2, M is the mass of the Earth
and r is the radius of the Earth (onto which a body
is falling).
Galileo (~1570 – 1642) derived the equation
F=-mg for bodies falling on the Earth, and found
g=980 cm/sec2.
Newton’s law is a generalization of Galileo’s
result, for bodies of any M and r.
G and M always appear together. To measure
M, one must know G.
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Newton had to show, mathematically, that for
perfect, uniform sphere, one can treat the
entire mass of the Earth as if it is a point
source at the center of the Earth. (For objects
like the Earth, this rule is a very good
approximation, even though the Earth is not
perfectly homogeneous.)
He had to assume that the nature of the
material of Earth did not affect the value of G
or the (1/r2) form of the force law.(No one has
ever been able to show this assumption to be
wrong.
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