Copyright © 2010 Pearson Education, Inc. Chapter 12: Gravitation

Copyright © 2010 Pearson Education, Inc.

Chapter 12:Gravitation


History of Gravitational Theory

• Ancient Greece & Ptolemy

• Nicolas Copernicus

• Tycho Brahe

• Johannes Kepler

• Isaac Newton

• Albert Einstein


Ancient Greece & Ptolemy

• Early theory had the Earth at the center of the universe.– “Geocentric”

• All stars were fixed at the same distance from the Earth.– Think of them on a sphere

surrounding Earth.• Saw objects that seemed

to “wander” through the night sky.


Ancient Greece & Ptolemy

• Aristotle & Plato said the planets circle perfectly around the Earth.

• 2nd century AD Ptolemy created a complicated system of circles and epicycles which could explain all types of movements.


Nicolas Copernicus (1473-1543)

• Born in Poland, 1473. Became a priest but maintained an interest in sky observations.

• Age 41, gave friends an anonymous manuscript arguing Ptolemy’s model would be greatly simplified if the sun were at the center of the universe.


Tycho Brahe (1546-1601)

• Danish noblemen born in 1546. Discovered a new star when he was 26, securing funding for the best astronomical observatory at the time.

• Rejected Copernican view of the universe and adopted his own. – Planets orbit the sun, which

orbits Earth the center.• Meticulous data

collection.


Johannes Kepler (1571-1630)

• Born in Germany, spent early years as a math teacher.

• Brahe needed Kepler’s mathematical prowess, Kepler needed Brahe’s data.

• After Brahe’s death, realized planets orbited in ellipse’s, not circles.

• Kepler’s Laws of Planetary Motion.


Sir Isaac Newton (1642-1727)

• Invented calculus.• Apple observation?• Law of Universal

Gravitation• Held the Lucasian

Chair of Mathematics at Cambridge University


Albert Einstein (1879-1955)

• Challenged Newton’s Theory of Gravity

• Special Theory of Relativity (1905)

• General Theory of Relativity (1915)– Gravity bends the

space around objects.


12-1 Newton’s Law of Universal Gravitation

Newton’s insight:

The force accelerating an apple downward is the same force that keeps the Moon in its orbit.

Hence, Universal Gravitation.



The gravitational force is always attractive, and points along the line connecting the two masses:

The two forces shown are an action-reaction pair.



The radius term in Newton’s Law of Universal Gravitation is the center-to-center distance of the two objects.


Exercise 12-1A man takes his dog for a walk on a deserted beach. Treating people and dogs as

point objects for the moment, find the force of gravity between the 105-kg man and his 11.2-kg dog when they are separated by a distance of (a) 1.00 m and (b) 10.0 m.


Question 12.1b Earth and Moon II

a) one quartera) one quarter

b) one halfb) one half

c) the samec) the same

d) two timesd) two times

e) four timese) four times

If the distance to the Moon were If the distance to the Moon were

doubled, then the force of doubled, then the force of

attraction between Earth and the attraction between Earth and the

Moon would be:Moon would be:



G is a very small number; this means that the force of gravity is negligible unless there is a very large mass involved (such as the Earth).

If an object is being acted upon by several different gravitational forces, the net force on it is the vector sum of the individual forces.

This is called the principle of superposition.


Force of Gravity During a Lunar Eclipse


12-2 Gravitational Attraction of Spherical Bodies

Gravitational force between a point mass and a sphere: the force is the same as if all the mass of the sphere were concentrated at its center.



What about the gravitational force on objects at the surface of the Earth? The center of the Earth is one Earth radius away, so this is the distance we use:

Therefore,



The acceleration of gravity decreases slowly with altitude:


Example 12-2If you climb to the top of Mt. Everest, you will be about 8.86

km above sea level. What is the acceleration due to gravity at this altitude?



Once the altitude becomes comparable to the radius of the Earth, the decrease in the acceleration of gravity is much larger:



The Cavendish experiment allows us to measure the universal gravitation constant:



Even though the gravitational force is very small, the mirror allows measurement of tiny deflections.

Measuring G also allowed the mass of the Earth to be calculated, as the local acceleration of gravity and the radius of the Earth were known.


12-3 Kepler’s Laws of Orbital Motion

Johannes Kepler made detailed studies of the apparent motions of the planets over many years, and was able to formulate three empirical laws:

1. Planets follow elliptical orbits, with the Sun at one focus of the ellipse.


Anatomy of an Ellipse

PerihelionAphelion

A

B

Focus

C


Anatomy of an Ellipse• Perihelion (or perigee): point in the orbit of a

planet, asteroid or comet where it is nearest to the sun.

• Apehelion (or apogee): point in the orbit of a planet, asteroid or comet where it is farthest from the sun.

• Semi-Major Axis (A): farthest distance from ellipsoid center to the edge of the ellipsoid.

• Semi-Minor Axis (B): shortest distance from ellipsoid center to the edge of the ellipsoid.



2. As a planet moves in its orbit, it sweeps out an equal amount of area in an equal amount of time.


The Earth’s orbit is slightly elliptical. In fact, the Earth is closer to the Sun during the (northern hemisphere’s) winter than it is during the summer. Is the speed of the Earth during winter (a) greater than, (b) less than, or (c) the same as its speed during the summer?


Conservation of Angular Momentum



3. The period, T, of a planet increases as its mean distance from the Sun, r, raised to the 3/2 power.

This can be shown to be a consequence of the inverse square form of the gravitational force.


Derivation of Kepler’s Third Law



A geosynchronous satellite is one whose orbital period is equal to one day. If such a satellite is orbiting above the equator, it will be in a fixed position with respect to the ground.

These satellites are used for communications and and weather forecasting.


The National Weather Service wants to put a new geosyncrhonous weather satellite into low Earth orbit. Find the altitude above the Earth’s surface where a satellite orbits with a period of one day.



GPS satellites are not in geosynchronous orbits; their orbit period is 12 hours. Triangulation of signals from several satellites allows precise location of objects on Earth.



Kepler’s laws also give us an insight into possible orbital maneuvers.

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Copyright © 2010 Pearson Education, Inc. Chapter 12: Gravitation