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Physics 55Friday, September 23, 2005

- Keplers empirical laws of planetary motion
- Newtons laws of motion and related concepts such

as mass, acceleration, and forces.

Need to Understand Some Physics of Gravity and

Light

- To make further progress in understanding

astronomy, you need to know some basic physics

concepts - Newtons laws mass, force, and universal law of

gravitation. - Fundamental conservation laws of energy,

momentum, angular momentum. - What we can learn from light surface

temperature, speed toward or away, rotation rate,

presence of atmosphere, atomic composition,

presence and strength of electric or magnetic

fields,

Calculating Periods

- Planets motion in sky results from combination

of true motion and Earths motion. Planet orbits

Sun each sidereal period. - What we see recurs every synodic period (relative

configuration of Earth, Sun, planet). - For an inferior planet, a synodic period has

elapsed when the planet has lapped Earth. - For a superior planet when Earth has lapped it.

- If sidereal period is P days, planet moves 360/P

degrees a day. Earth moves 360/E degrees a day

where E365 is sidereal period of Earth. - Planet laps Earth (completing a synodic period of

S days) when S(360/P) S(360/E) 360. This is

same as 1/P 1/E 1/S. - For a superior planet find similarly 1/P 1/E

1/S.

Galileos Smoking Scope

- Smoking gun evidence for Copernician model

required new technology. - Galileo (1610) turns new telescope up and finds

phases of Mercury. - The correlation between phase and position in sky

agrees with heliocentric model, not with

Ptolemaic model.

Cultural Issues

- Galileo also discovers that Jupiter is itself

accompanied by moons that orbit the planet, much

like our Moon orbits Earth. Nature repeats on

different scales. - Motion of Jupiters moons studied closely, forms

first Nautical clock for longitude measurement. - Despite all this, Galileo tried for heresy and

sentenced to house arrest.

More on Galileo

- Galileo made many other discoveries of

importance. With his telescope, he discovered

mountains on the moon, size and shape of planets,

nature of Milky Way, moving spots on Sun, among

others. - He also studied mechanics, properties of motion

in general. Formulated principle of inertia

object tends to remain in its state of motion

unless disturbed externally. We know we need to

work to move things. Galileos insight reminds us

we also work to stop or turn them. - Galileo almost got mechanics right. What stopped

him was the fact that the mathematics needed to

formulate the theory was not known. To make

progress, Newton had to invent Calculus.

Brahe and Kepler

- First steps to deeper insight were careful

observations by Brahe (1580) of planetary motion

to great precision. - Using these, Kepler (1609) finds three laws of

planetary motion.

Keplers Laws

- 1. Orbit of a planet is an ellipse with Sun at

one focus. - Ellipse is shape of all points such that sum of

their distances from two points (foci) is

constant. - Eccentricity (e) measures how far the foci are

relative to size. e0 is a circle. - Other focus is nothing (not even same for all

planets). - Typically e small, .017 for Earth, .2 for Mercury.

- 2. Line connecting planet to Sun sweeps out

equal areas in equal time intervals as planet

orbits. - Planet moves faster at perihelion, slower at

aphelion. - This causes slight change in rate of Suns motion

discussed earlier.

- This effect much more dramatic for comets which

follow highly eccentric orbits.

- 3. Square of sidereal period is proportional to

cube of semimajor axis. - This relates the orbital motions of different

planets orbiting same Sun. - Write this as P2 a3. This is valid if P is

measured in years and a in AU.

- Recall 1AU 1.496 108 km 93 million miles is

average Earth-Sun distance.

- Keplers laws are amazing progress. They give

planetary motion with unprecedented accuracy.

Whats more, they are universal they apply to

any orbital system, from an atom through Saturns

moons to Galaxy clusters. - In physics such universality means there are

fundamental laws at work here. - These were found by Newton (1670) who at first

was not thinking at all about Astronomy.

PRS Question

- An asteroid with an orbital period of 8 years

lies at - an average distance from the Sun equal to
- 2 AU
- 4 AU
- 8 AU
- 16 AU
- Need to know the asteroids mass.

PRS Question

- The period of revolution p of a point on a

spinning - CD is related to its distance r from the center

of the - CD (its axis of rotation) by the expression
- p2 c r3 for some constant c.
- p2 c r for some constant c.
- p c r for some constant c.
- p does not depend on r.

PRS Question Prediction

If steel ball is being swung around in circle on

a rope and if the rope breaks at point P, which

path does ball follow next? (For PRS, A1, B2,

etc.)

PRS Question Prediction

PRS Question Prediction

Newtons Laws of Motion

- 1. An object upon which no forces act will move

in a straight line with constant velocity.

(inertia)

- Familiar when velocity is zero object at rest

will stay there.

- Need to remember in our world two forces (at

least) always get in the way gravity pulls us

down friction slows all motion. To see Newton 1

need to minimize these or imagine them removed.

- Velocity is a vector has direction as well as

magnitude. So constant velocity means no change

in direction or speed.

- 2. When a force acts on an object, it will

change its velocity. The acceleration will be

proportional to the force (and pointed in the

same direction). The proportionality constant is

called mass. - F ma

- Acceleration a is rate of change of velocity v.

So measured in (m/sec)/sec or m/sec2. - Like v it is a vector and has direction. Note

that changing direction of v requires

acceleration, just as does changing magnitude of

v.

- m is mass. Measured in kg total amount of

stuff.

- F is force. Measured in kg (m/sec2) N(ewton)

- Acceleration of gravity here is g 9.8 m/sec2,

so force of gravity on 1kg. is 9.8 N.

My Van

- My van can go 0 to 60 mph in 12 sec.
- This is an acceleration of
- a (60 mi/hr)/12 sec
- 5 (mi/hr)/sec
- (5 1609 m/3600 sec)/sec
- 8045 m/3600 sec2
- 2.34 m/sec2
- Its mass is 800 kg. Force required is
- F ma 800 kg2.34 m/sec2
- 1788 kgm/sec2 1788 N

a (1) 5 mph/sec (2) 5 m/sec2 (3) 5

miles/sec2

F (1) 1743 m/sec2 (2) 1788 N (3)

1967 N

Uniform Circular Motion

- Planet moves at uniform speed v around circle of

radius R. Period is P2pR/v - Is velocity constant?

- NO. Direction changes.

- Guess acceleration
- Points inwards
- Grows with larger v (m/sec).
- Smaller with larger R (m).
- Measured in m/sec2.

a (1) v/ R2 (2) v2 R (3) v2 / R

- a v2/R.
- So F ma m v2/R

Numbers for Earth

- As Earth spins, we move at
- Vspin (2pR)/P 4107 m/24 hr
- 463 m/sec 1036 mph
- As Earth orbits, we move at
- Vorbit (2pR)/P
- 6.28 1.5 1011 m/36586400 sec.
- 29871 m/sec 100,595 mph

- a v2/R
- 4632/6.38106
- 0.034 m/sec2

a (1) 9.8 m/sec2 (2) 0.034 m/sec2

(3) 0.45 m/sec2

- F ma
- 100kg. 0.034m/sec2
- 3.4 N

- a v2/R
- 298712/1.51011
- 0.0059 m/sec2

- F ma
- 100kg. 0.0059m/sec2
- 0.59 N

- When rocks in space hit Earth the relative

velocities are about 100,000 mph. That is why

they burn in atmosphere as meteors!

- Lets compute the forces required to keep in

these circular motions a person of mass m100 kg.

He weighs 9.8 m/sec2 100 kg 980 N

- 3. When one object applies a force to another,

the latter applies a force to the former, equal

in magnitude and opposite in direction. (action

and reaction). - This explains how we walk. I push Earth back, it

pushes me forward!

Deductions by Newton

- Elliptical orbit suggested to Newton an

inverse-square law for gravity. - Keplers first law was almost but not exactly

correct ellipses are the true shape of an orbit

only for two isolated masses. Can deduce

position, mass of unknown planets from tiny

deviation of known planet from ellipse. - Keplers second law holds for any central force,

is really a statement about conservation of

angular momentum. - Keplers third law can be generalized to a more

useful form that allows one to deduce the mass of

the less massive object in orbit. - Two masses orbit around their center of mass,

which is at a focus of their elliptical orbits.

Important for binary stars, Pluto and its moon

Charon. - Other conic sections such as parabolas and

hyperbolas can describe unbounded orbits of one

mass moving near a second mass. - Escape velocity, implication for black holes.
- Tidal stresses, origins of tides, Roche limit,

black holes.

Brief Review at Whiteboard of Acceleration, Mass,

Forces, Gravity

Logic Objects like balls and planets often have

nonuniform motion called acceleration.

Acceleration has physics units speed over time or

m/s2. Experiments and thinking suggested to

Newton that acceleration can only arise from

something called a force, which acts on a body to

change its speed or direction. Many kinds of

forces gravitational, electrical, magnetic,

friction. Acceleration a is related to force F

by a positive quantity called the mass m of the

object

a (1/m)F. The bigger the

mass, the smaller the acceleration for a given

force. Note mass is measured in units of

kilograms kg force is measured in units of mass

x acceleration kg m/s2 called a newton and

abbreviated as N. By brilliant mathematical and

scientific thinking, Newton discovered a formula

for the gravitational force F of one mass on

another mass. Newton also realized that the

formula is universal and applies to any two

masses in space apples, Moons, planets, stars.

Simplest Motion Uniform Motion

Nonuniform Motion Acceleration

Speed is not constant or direction of motion is

not constant (but speed can be constant in case

of circular motion).

A

Where are speeds large in this picture if

stroboscope samples at equal times? Where are

speeds small in this picture?

B

Demo Ball in Circular Motion

Ball would go in straight line (B) if a force

didnt act on the ball, which here is the string

pulling the ball toward the person at the center

of the circle.

PRS Question

Newtons Great InsightNonuniform Motion Caused

by Forces

Something from one object like Sun somehow

influences motion of other object like Earth.

That something is still not understood in any

fundamental sense but Newton discovered could be

described by an astonishingly simple and precise

mathematical rule now known as the universal law

of gravitation. The gravitational force becomes

weaker with distance but has an effect no matter

how far one object is from other object.. Total

force on object is sum of forces from all other

objects so depends on relative positions of all

the other objects. Mathematics can be hard,

computers have helped to obtain insight.

Orbit Falling Around the Earth

Marvelous insight and calculation by Newton Moon

falls around Earth exactly as apple falls to the

ground, gravity is quantitatively universal.

How to FlyThe Hitchhikers Guide to the Galaxy

There is an art, it says, or rather, a knack to

flying.The knack lies in learning how to throw

yourself at the ground and miss. Pick a nice day,

it suggests, and try it.The first part is easy.

All it requires is simply the ability to throw

yourself forward with all your weight, and

willingness not to mind that it's going to hurt.

That is, it's going to hurt if you fail to miss

the ground. Most people fail to miss the ground,

and if they are really trying properly, the

likelihood is that they will fail to miss it

fairly hard.