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Summary

Selected Vocabulary

ellipse................................a flattened circle; one of the conic sections

conic section......................a curve formed by the intersection of a plane and an infinite cone

hyperbola..........................another conic section; it does not close back on itself

period................................the time required for a planet to complete one orbit; more generally, the

time for one repetition of some repeating motion

focus..................................one of two special points inside an ellipse: the ellipse consists of all points

such that the sum of the distances to the two foci equals a certain number;

a hyperbola also has a focus

Notation

G.......................................the constant of proportionality in Newton’s law of gravity; the gravita-

tional force of attraction between two 1-kg spheres at a center-to-center

distance of 1 m

Summary

Kepler deduced three empirical laws from data on the motion of the planets:

Kepler’s elliptical orbit law: The planets orbit the sun in elliptical orbits with the sun at one focus.

Kepler’s equal-area law: The line connecting a planet to the sun sweeps out equal areas in equal

amounts of time.

Kepler’s law of periods: The time required for a planet to orbit the sun is proportional to the long axis

of the ellipse raised to the 3/2 power. The constant of proportionality is the same for all the planets.

Newton was able to find a more fundamental explanation for these laws. Newton’s law of gravity states

that the magnitude of the attractive force between any two objects in the universe is given by

F = Gm

1

m

2

/r

2

.

Weightlessness of objects in orbit around the earth is only apparent. An astronaut inside a spaceship is

simply falling along with the spaceship. Since the spaceship is falling out from under the astronaut, it appears

as though there was no gravity accelerating the astronaut down toward the deck.

Gravitational forces, like all other forces, add like vectors. A gravitational force such as we ordinarily feel is

the vector sum of all the forces exerted by all the parts of the earth. As a consequence of this, Newton proved

the shell theorem for gravitational forces:

If an object lies outside a thin, uniform shell of mass, then the vector sum of all the gravitational forces

exerted by all the parts of the shell is the same as if all the shell’s mass was concentrated at its center.

If the object lies inside the shell, then all the gravitational forces cancel out exactly.

Chapter 10Gravity