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Unformatted text preview: Chapter 8 Lecture Essential University Physics Richard Welfson
2nd Edition Gravity Universal Gravitation  Introduced by Isaac Newton, the Law of Universal
Gravitation states that any two masses mI and m2 attract with a force F that is proportional to the product of their distances
and inversely proportional to the distance r between them:  Here G = 6.67x101" NmEIkg2 is the constant of universal
gravitation.  Strictly speaking, this law applies only to point masses. But
Newton showed that it applies to spherical masses of any size,
and it is a good approximation for any objects that are small
compared with their separation. Circular Orbits  In a circular orbit, gravity provides the force of magnitude mv2ir needed to keep an object of mass min
its circular path about a much more massive object of mass M. Therefore, GMm _ W2 Fl 3“  Orbital speed: 12:1/GM r . . 2 3
 Orbital period: T2 :4” 3”
GM — Kepler’s third law: T2 pp 1‘3 — For satellites in lowEarth orbit, the period is about 90
minutes. Gravitational Potential Energy  Because the gravitational force changes with distance, it’s necessary to integrate to calculate potential energy
changes over large distances. This integration gives r2 r2 _1 a_1 Q 1 1
AUIE :I_ mdFIGMHT i" ‘di“:GMmL r GMm[__] 2 1
r _ x.
r 1 1 :1 f2 whether the two points are on
the same radial line. ' " 1, Hiiiuc hillllllLlL‘ tlLiLHii'l  It’s convenient to take the zero of vliullvc — H
. . . till i 'IIH [Till ]  . 
gravitational potential energy at =  This Fesult holds regardless of O
M infinity. Then the gravitational 1. — H I
 'f...._.. .  . m it." a is
potential energy becomes M H.
GM”? xi':*.~»l;1il]icic. Um : — F. Energy and Orbits  The total energy E = K + U, the sum of kinetic energy K and potential energy U, determines the type of orbit an
object follows: — E < O: The object is in a bound, elliptical orbit.  Special cases include circular orbits and the straight
line paths of falling objects. — E > 0: The orbit is unbound and hyperbolic.
— E = O: The borderline case gives a parabolic orbit. "rrl ii maximum LlihlllilL'IiL‘ with energy LI. I Gravitational potential energy. if Escape Speed  An object with total energy E less than zero is in a bound
orbit and can’t escape from the gravitating center.  With energy E greater than zero, the object is in an unbound
orbit and can escape to infiniter far from the gravitating center.  The minimum speed required to escape is given by a GM
0:K+U:§mw——m
I.  Solving for vgives the escape speed: i 2 GM
VESC :
I, — Escape speed from Earth’s surface is about 11 kas. Energy in Circular Orbits  In the special case of a circular orbit, kinetic energy and
potential energy are precisely related: biz—2K
 Thus in a circular orbit the total energy is
E:K+U:—K:%U:—€?m
F. — This negative energy shows that the orbit is bound. — The lower the orbit, the lower the total energy—but the
faster the orbital speed.  This means an orbiting spacecraft needs to lose energy to ain s eed. is] lbl Energy and Orbits  The total energy E = K + U, the sum of kinetic energy K and potential energy U, determines the type of orbit an
object follows: — E < O: The object is in a bound, elliptical orbit.  Special cases include circular orbits and the straight
line paths of falling objects. — E > 0: The orbit is unbound and hyperbolic.
— E = O: The borderline case gives a parabolic orbit. "rrl ii maximum LlihlllilL'IiL‘ with energy LI. I Gravitational potential energy. if ...
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This note was uploaded on 02/12/2012 for the course PHYSICS 104 taught by Professor Staff during the Fall '10 term at Rutgers.
 Fall '10
 Staff
 Physics

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