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Physics 121C - MechanicsLecture 19Universal

GravitationNovember 19, 2004

- John G. Cramer
- Professor of Physics
- B451 PAB
- cramer_at_phys.washington.edu

Announcements

- Regrade requests for Exam 2 will be accepted

through noon on Monday, November 22. - Homework Assignment 6 has been posted on Tycho

and is due on Wednesday, November 24.

Lecture Schedule (Part 3)

You are here!

The Pre-History of Gravitation

The ancients observed that the stars were

fixed, while the planets moved against the

background of fixed stars. They were very

interested in the stars because the movements of

the stars were correlated with the seasons,

growing cycles, etc. The discipline of astrology

also asserted that the movements of the planets

influenced the lives and destinies of humans, and

that future events could be predicted by studying

and codifying planetary movements. This created

an industry for those inclined to learning and

skilled in calculation and geometry.

Aristotle (384 BC - 322 BC)

The Greek philosopher Aristotle taught that

the Earth was at the center of a nested set of

transparent spheres, with the fixed stars onthe

outer sphere andthe planets (includingthe Sun

and Moon)attached to innerspheres, all

rotatingat differing rates.

Claudius Ptolemy (85-165)

Claudius Ptolemy (2nd century AD) noted that

some of the planets showed retrograde motion,

appearing to reverse direction as they moved

against the stars, in seeming contradiction to

Aristotles model of celestial spheres.

Ptolemy explained this by attaching the planets

to sub-spheres that rotated on the main spheres,

so that planetary motion was described by nested

epicycles.

Ptolemys cosmology became the Standard

Model of the universe for about 1,400 years.

Nicolaus Copernicus (1473-1543)

Copernicus, in his book De

Revolutionibus(published posthumously) argued

that the Sun wasthe center of the universe, and

that the Earth wasone of the planets revolved

about it in circular orbits.The rationale for

the circles was in part theological, circles

being perfect geometrical objects. The

Church banned Copernicus book andpersecuted

those who accepted his ideas, becausehis

assertions were in conflict with the

foundationsof medieval theology. From 1570 to

1600, the Danish astronomer Tycho Brahe compiled

a set of extremely accurate (pre-telescopic)

astronomical observations. Tychos observations

revealed that there were problems with

Copernicus assertion that the planet followed

circular orbits.

Johannes Kepler (1571-1630)

- Johannes Kepler inherited Tychos

observations and tried to makesense of them,

using algebra, trigonometry, and geometry. After

a decadeof work, he was forced to conclude that

planetary orbits were betterdescribed by

ellipses than by circles, and that the planets

travel in theseorbits with avarying speed. He

deduced three laws of planetary motion - All planets move in in elliptical orbits, with

the Sun as a focus ofthe ellipse. - A line drawn between Sun and planet sweeps out

equal areas inequal times. - The square of a planets orbit period is

proportional to the cubeof the length of its

semi-major axis.

Galileo Galilei (1564 -1642)

In Pisa, Italy, Galileo Galilei hear rumors

fromvisiting sailors of a device invented in

Holland thatallowed one to obtain a magnified

view of distantobjects. He experimented with

lenses until he re-discovered the trick, which

was placing a strongdiverging lens near the eye

while viewing a weakerand larger converging lens

places further away.He discovered (or

re-discovered) the telescope. He used this

invention to view the stars and planets.He

discovered that the planet Venus has phases, like

theMoon, that Saturn had rings, and that four

tiny points oflight can be seen around Jupiter.

These moons of Jupiter formed a miniature solar

system, demonstrating the validity of the ideas

of Copernicus and Kepler. Galileo published

his observations and ideas, and he was arrested

by the Inquisition. He was tried and convicted

of heresy and was forced to publicly recant his

views.

Isaac Newton (1642 - 1727)

Isaac Newton was born in 1642, the year of

Galileosdeath. He entered Trinity College of

Cambridge Universityat the age of 19 and

graduated in 1665, at the age of 23.Because the

Black Death was ravaging Europe at the time,he

returned to his family farm for two years to

escape thepestilence. It was during this

period that he did his greatest work.He

performed experiments in optics, laid the

foundationsof theories of mechanics and

gravitation, and, because heneeded it, invented

calculus as a new branch ofmathematics.

Newton, following an idea suggested by Robert

Hooke, hypothesized that the force of gravity

acting on the planets is inversely proportional

to their distances from the Sun.

The Appleand the Moon

The radius of the Moons orbit is

RM3.84x108 m. If T 2pr/g½ and g9.80 m/s2,

then the Moons orbital period should be TM

2pRM/g½ 2p(3.84x108 m)/(9.80 m/s2)½

3.93x104 s 11 hr. However, the actual

orbital period of the Moon is about 27.3 days

2.36x106 s. How could this calculation be so

badly off? Lets use the Moons orbital

period and calculate gM, the acceleration due to

Earths gravity at the orbit of the Moon.gM

RM(2p/T)2 (3.84x108 m)2p/(2.36x106 s)2

2.72x10-3 m/s2 But an apple falls at gE9.80

m/s2. Lets try something. Well calculate the

product gR2 for an apple at the Earths surface

and for the Moon in orbit

gMRM2(2.72x10-3 m/s2)(3.84x108 m)24.01x1014

m3/s2 gERE2 (9.80 m/s2)(6.37x106 m)2

3.98x1014 m3/s2 These products are

essentially equal, because gravity falls off

1/R2. The same gravity affects the apple and the

Moon.

Newtons Law of Gravity

- Newton proposed that every object in the universe

attracts every other object with a force that has

the following properties - The force is inversely proportional to the

distance between the objects. - The force is directly proportional to the product

of the masses of the two objects.

Gravitational Force and Weight

With Newtons Law of Gravity, we can

calculate the gravitational force produced by the

Earth and acting on some mass on the Earths

surface. (To do this, we assume that the Earths

gravity is that same as it would be if all of the

Earths mass were concentrated at its center.)

Gravity is a very weak force, much weaker

than the other three forces of nature (the

strong, electromagnetic, and weak interactions).

However, it is a long-range force and it is

cumulative. It always adds, never subtracts,

because there is no (known) negative mass in the

universe.

The Principle of Equivalence

- Mass appears in two roles in physics
- Inertial mass, which resists acceleration
- Gravitational mass, which produces gravitational

attraction.

The Principle of Equivalence states that these

masses are always equal, and that the apparent

force in an accelerated reference frame is

indistinguishable from gravity.

The Principle of Equivalence

Little g and Big G

On other planets the acceleration due to

gravity (gX) will be different, because it

depends on the mass and radius of each planet.

However, the law of gravity is universal, so a

physicist on Planet X would measure the same

value for G that we measure on Earth.

Rotation and Little g

Notice that we calculated a value for g that

was slightly larger than 9.80 m/s2. This is

because the Earth is rotating, and part of the

force of gravitational attraction acts to provide

centripetal acceleration, keeping objects moving

in a circular path as the Earth rotates. The

centripetal acceleration is about 0.03 m/s2,

accounting for the difference.

Decrease of g with Distance

Weighing the Earth

Newtons gravitational constant G must be

measured in the laboratory. Henry Cavendish made

the first accurate measurement of this quantity,

using a Cavendish balance. The forces between

masses are measure using their action in twisting

a thin fiber. G is calculated from the measured

force.

A measurement of G is essentially a

measurement of the mass of the Earth.

Gravitational Potential Energy (1)

So far, we have used Ugmgy for the

gravitational potential energy, where y is the

height above the surface of the Earth. Now we

would like to do better, using the law of gravity.

We consider a mass m2moving in the gravity

of mass m1 from some radius r to infinity. This

is the potential energy, with DU0 at infinity

where the force goes to zero.

Gravitational Potential Energy (2)

Example Crashing into the Sun

Suppose the Earth were suddenly to halt its

motion in orbit the Sun. The gravitational force

would pull it directly into the Sun. What would

be its speed as it crashed?

Example Escape Speed

A 1000 kg rocket is fired straight away from

the surface of the Earth. What speed does it

need to escape from the gravitational pull of

the Earth and never return? (Assume a

non-rotating Earth.)

The Flat-Earth Approximation

This is sometimes called the Flat Earth

Approximation. It is consistent with our

previous treatment of gravitational potential

energy.

ExampleThe Speed of a Projectile

- A projectile is launched straight up from

the Earths surface. - With what speed should it be launched if it is

to have a speed of 500 m/s at a height of 400 km? - By what percentage would your answer be in error

if you use the Flat-Earth approximation?

This is too big by 2.5.

Clicker Question 1

Which of these systems has the largest

absolute value of gravitational potential energy

Ug ?

End of Lecture 19

- Before the next lecture, read Knight, Sections

12.6 through 13.2. - Homework Assignment 6 has been posted on Tycho

and is due on Wednesday, November 24. - Regrade requests for Exam 2 will be accepted

through noon on Monday, November 22.