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Physics 2211 - MechanicsLecture 18 (Knight

12.1 to 12.5)Universal Gravitation

- Dr. John Evans

Heat

So far, we have this relation for energy

conservation

This relation does not include the

possibility that energy is being added to the

system as thermal energy. To include this

possibility, we add the term Q to the equation,

representing external heating of the system. The

heat Q transfers energy to and from the system

when there is a temperature difference between

the system and the external environment. To

include this effect, we now write

Power

Work transfers energy between the

environment and the system. In many cases we are

interested in how fast this energy transfer

occurs. In other words, we want to consider the

rate of energy transfer, which is called power,

defined as P º dEsys/dt. The SI unit for

power is the watt (W), which is defined as 1 J/s.

Another common power unit is horsepower (hp),

defined as 745.6999 W.

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 doctrine of

astrology asserted thatthe 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 createdan industry for those inclined to

star observation, learning, and skill in

calculationand geometry. Thus superstition

drove science.

Aristotle (384 BC - 322 BC)

The Greek philosopher Aristotle taught that

the Earth was at the centerof a nested set of

transparent spheres,with the fixed starson the

outersphere and theplanets (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 Sunwas the center of the universe, and

that theEarth was one of the planets that

revolvedabout it in circular orbits. The

rationale forthe circles was in part

theological, circles beingperfect geometrical

objects. The Church banned Copernicus book

andpersecuted those who accepted his

ideas,because his assertions were in conflict

withthe foundations of 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 heard rumors

fromvisiting sailors of a device invented in

Holland thatallowed one to obtain magnified

views of distantobjects. He experimented with

lenses until he re-discovered the trick, which

was placing a strongdiverging lens near the eye

while viewing a throughweaker and larger

converging lens placed furtheraway. 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, likethe Moon, that Saturn had rings, and

that four tinypoints of light can be seen around

Jupiter. These moonsof Jupiter formed a

miniature solar system, demonstratingthe

validity of the ideas of Copernicus and Kepler.

Galileo published his observations and ideas,

and hewas arrested by the Inquisition. He was

tried andconvicted 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

CambridgeUniversity at the age of 19 and

graduated in 1665, atthe age of 23. Because the

Black Death was ravagingEurope at the time, he

then returned to his familysfarm estate for two

years to escape the pestilence. It was

during this period that he did his greatestwork.

He performed experiments in optics, laid

thefoundations of his theories of mechanics

andgravitation. Because he needed it for his

studies, heinvented the calculus as a new branch

of mathematics. 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. This is now called Newtons Law of

Gravity.

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 3.93

x 104 s 11 hr. However, the actual orbital

period of the Moon is about 27.3 days 2.36 x

106 s. How could this calculation be so badly

off? (Weaker gravity?) 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 2.72 x 10-3 m/s2 But an apple falls

at gE 9.80 m/s2. So 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)2

4.01x1014 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 gravitational

force law 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

an alien 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 Ug mgy 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 m2 moving in the gravity

of mass m1 from some radius r to infinity. This

is the potential energy, with DU 0 at infinity

where the force goes to zero.

Gravitational Potential Energy (2)

A plot of the gravitational potential energy

Ug looks like this

Example Crashing into the Sun

Suppose the Earth were suddenly to halt its

motion in orbiting 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.)

This is also the speed at which (in the

absence of atmosphere) a meteor, falling from

very far away, would strike the surface of the

Earth. It is called escape velocity.

The Flat-Earth Approximation

This is sometimes called the Flat Earth

Approximation. It is consistent with our

previous treatment of gravitational potential

energy in Chapter 10.

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 18

- Before the next lecture, read Knight, Sections

13.1 through 13.3. - Uncollected exam papers are available from Laura

Clement, room C136 PAB. - Check Tycho to make sure you have a grade for

all parts of the exam. If not, see Laura. - Regrade requests for Exam 2 will be accepted

through noon on Monday, November 21. - Homework Assignment 7 is posted on Tycho and is

due by Midnight on Wednesday, November 23.

Satellite Orbits and Energies

The tangential velocity v needed for

acircular orbit depends on the

gravitationalpotential energy Ug of the

satellite at theradius of the orbit. The needed

tangentialvelocity v is independent of the mass

m ofthe satellite (provided mltltM).

ExampleThe Speed of the Space Shuttle

The Space Shuttle, in an orbit 300 km above

the surface of the Earth, wants to capture a

smaller satellite for repairs. What are the

speeds of the Shuttle and the satellite in this

orbit?

Keplers 3rd Law

Therefore, Keplers 3rd Law is a direct

consequence of Newtons Law of Gravity. In

the Log-Log plot to the right, the data for the

planets of the Solar System fall on a power-law

straight line specified by log10T 1.500

log10r - 9.264 The 2nd term can be used to

calculate the mass of the Sun.

The Solar System

Geosynchronous Orbit

In 1945 the science fiction author Arthur C.

Clarke pointed out that it was possible to put a

satellite in an orbit above the equator that had

a period of exactly one day, so that it rotated

around the Earth at the same rate that the Earth

rotated under it. Such a geosynchronous

satellite hangs above a particular point on the

equator and is now widely used for

communications. Clarke also envisioned

lowering a rope from a geosynchronous space

station and hauling objects into space without

rockets, using a space elevator. This is now

being seriously considered, using a super-strong

cable made from carbon nanotubes.

Notice this is the cube-root.

Example Extrasolar Planets

Astronomers, using the most advanced

telescopes, have recently began to discover

planets orbiting nearby stars, usually deduced

from a wobble in the stars position at the

orbital period of the planet. Suppose a

wobble with a 1200 day period (1.037x 108 s) is

observed, and it is assumed that the planet is

the same distance from its star that Jupiter is

from the Sun. What isthe mass of the star, in

solar masses?

Keplers 2nd Law

Keplers 2nd Law is a consequence of the

conservation of angular momentum.

Kepler vs. Newton

Are Keplers Laws really laws, in the

sense of Newtons Laws? No. Keplers Laws

are empirical rules deduced from data, and are

approximate, because they include only the

gravitational interaction between each planet and

the Sun, while ignoring the mutual gravitational

interactions between the planets. Newtons

Laws, on the other hand, are true Laws of Nature

that allow us to deduce all of the forces acting

in the Solar System, including planet-planet

interactions, and to calculate and predict orbits

to whatever precision we desire. We note,

however, that Newtons Laws are also

approximations, because they do not include the

effects of special and general relativity, e.g.,

relativistic mass-increase at high velocities and

time dilation in strong gravitational fields.

Orbital Energetics

The equation K -½Ug is called The Virial

Theorem. In effect, it says that for a planet

in orbit around the Sun, if you turned its

velocity by 90o, so that it pointed straight out

of the Solar System, you would have only half the

kinetic energy needed to escape the Suns gravity

well.

ExampleRaising a Satellite - LEO to Geo

How much work must be done in boosting a

1000 kg communication satellite from low Earth

orbit (h300 km) to geosynchronous orbit?

Chapter 12 Summary (1)

GENERAL PRINCIPLES

Chapter 12 Summary (2)

Chapter 12 Summary (3)