Chapter 8 Review

Chapter 8 Review - Chapter 8 Lecture Essential University...

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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 = 6.67x10-1" N-mEIkg2 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 low-Earth 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‘ tlLiL-Hii'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 i-i 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 i-i 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.

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Chapter 8 Review - Chapter 8 Lecture Essential University...

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