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PHYS 218 sec Review Chap. 12 Gravitation

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What you have to know Newton’s law of gravitation Gravitational potential energy Motion of satellites Kepler’s three laws We skip from section 12.6 through section 12.8

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N. CopernicusG. Galilei T. Brahe J. Kepler I. Newton

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Newton’s law of gravity Always attractive In vector form, the gravitational force exerted by m 2 on m 1 is The negative sign means that it is an attractive force.

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Gravity for spherically symmetric bodies For an object which has spherically symmetric mass distribution: concentrate all the mass of the object at its center. Earth of mass m E

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Ex 12.2 Acceleration due to gravity

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Ex 12.3 Superposition of gravitational forces Gravitational force is a vector. The gravitation force exerted on m = vector sum of two forces

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Weight Weight of a body: the total gravitational force exerted on the body by ALL other bodies in the universe At the surface of the Earth, we can neglect other stellar objects. Radius of the Earth Mass of the Earth

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Ex 12.4 Gravity on Mars Use this information to know the mass of the Mars lander At d = m above the surface of Mars

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Gravitational potential energy When the gravitational acceleration is constant In general, the gravitational acceleration depends on r Gravitational force displacement

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Gravitational potential energy II Gravitational force is conservative At the surface of the Earth = constant, so can be dropped

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Ex 12.5 From the earth to the moon Muzzle speed needed to shoot the shell from R E to 2R E To obtain the speed, we use energy conservation.

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Ex 12.5b From the earth to the infinity Muzzle speed needed to shoot the shell from R E to infinity This is called the escape speed Independent of the mass of the object

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Motion of satellites Closed orbits Open orbits

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Satellites: circular orbits The radius of the circular orbit of the satellite is determined by its speed. Independent of the satellite mass

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Satellites: circular orbits For a given radius, satellite speed is determined, so is its energy

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Ex 12.6 From the earth to the infinity Speed, period, acceleration

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Ex 12.6 Cont’d The work needed to place this satellite in orbit The additional work to make this satellite escape the earth

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Kepler’s laws Kepler’s First Law: each planet moves in an elliptical orbit, with the sun at one focus of the ellipse This can be shown by solving the equation of motion based on Newton’s theory on gravity and Newton’s second law of motion. (but needs higher level of math) Perihelion: distance between P and S is minimum. Aphelion: distance between P and S is maximum. e: eccentricity in most cases, e is very small and the orbit is close to a circle

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Kepler’s Second Law: A line from the sun to a given planet sweeps out equal area in equal times The line SP sweeps out equal areas in equal times A result of angular momentum conservation See the textbook for the proof. Kepler’s Third Law: The period of the planets are proportional to the 3/2 powers of the major axis lengths of their orbits We have seen this for the case of circular orbit. But this is true even for elliptic orbits.

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