Review Chap. 12 Gravitation PHYS 218 sec. 517-520 Review Chap. 12 Gravitation

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

N. Copernicus G. Galilei I. Newton J. Kepler T. Brahe

Newton’s law of gravity Always attractive In vector form, the gravitational force exerted by m2 on m1 is The negative sign means that it is an attractive force.

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 mE

Acceleration due to gravity Ex 12.2 Acceleration due to gravity

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

At the surface of the Earth, we can neglect other stellar objects. 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. Mass of the Earth Radius of the Earth

Use this information to know the mass of the Mars lander Ex 12.4 Gravity on Mars Use this information to know the mass of the Mars lander At d = 6000000 m above the surface of Mars

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

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

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

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

Motion of satellites Closed orbits Open orbits

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

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

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

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

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) e: eccentricity in most cases, e is very small and the orbit is close to a circle Aphelion: distance between P and S is maximum. Perihelion: distance between P and S is minimum.

A result of angular momentum conservation Kepler’s Second Law: A line from the sun to a given planet sweeps out equal area in equal times A result of angular momentum conservation The line SP sweeps out equal areas in equal times 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.