Gravitation.ppt

7/27/2019 Gravitation.ppt

1/19

Gravity

Newtons Law of Gravitation

Keplers Laws of PlanetaryMotion

Gravitational Fields


2/19

Newtons Law of Gravitation

m1

m2r

There is a force of gravity between any pair of objects

anywhere. The force is proportional to each mass andinversely proportional to the square of the distance between

the two objects. Its equation is:

FG =G m1m2

r2

The constant of proportionality is G, the universal gravitation

constant. G = 6.67 10-11 Nm2 / kg2. Note how the units of G all

cancel out except for the Newtons, which is the unit needed on the

left side of the equation.


3/19

Gravity Example

FG =G m1m2

r2

How hard do two planets pull on each other if their masses are

1.23

1026

kg and 5.21

1022

kg and they 230 million kilometersapart?

This is the force each planet exerts on the other. Note the denominator

is has a factor of 103 to convert to meters and a factor of 106 to

account for the million. It doesnt matter which way or how fast the

planets are moving.

(6.67 10-11Nm2 / kg2) (1.23 1026 kg) (5.21 1022 kg)=(230 103 106 m)2

= 8.08 1015 N


4/19

3rd Law: Action-Reaction

In the last example the force on each planet is the same. This is due to

to Newtons third law of motion: the force on Planet 1 due to Planet 2is just as strong but in the opposite direction as the force on Planet 2

due to Planet 1. The effects of these forces are not the same, however,

since the planets have different masses.

For the big planet: a = (8.08 1015 N) / (1.23 1026 kg)= 6.57 10-11 m/s2.

For the little planet: a = (8.08 1015 N) / (5.21 1022 kg)= 1.55 10-7 m/s2.

5.21

1022 kg1.23 1026 kg

8.08 1015 N 8.08 1015 N


5/19

Inverse Square Law

FG =G m1m2

r2

The law of gravitation is called an inverse square law because the

magnitude of the force is inversely proportional to the square of theseparation. If the masses are moved twice as far apart, the force of

gravity between is cut by a factor of four. Triple the separation and

the force is nine times weaker.

What if each mass and the separation were all quadrupled?

answer: no change in the force


6/19

Falling Around

the Earth v

Newton imagined a cannon ball fired

horizontally from a mountain top at a speed

v. In a time t it falls a distance y = 0.5gt2 while

moving horizontally a distancex = vt. If fired fast

enough (about 8 km/s), the Earth would curve downward

the same amount the cannon ball falls downward. Thus, the

projectile would never hit the ground, and it would be in orbit.The moon falls around Earth in the exact same way but at a

much greater altitude. .

x = vt

y = 0.5gt2 {

continued on next slide


7/19

Necessary Launch Speed for OrbitR= Earths radius

t= small amount of time after launch

x= horiz. distance traveled in time ty= vertical distance fallen in time t

RR

y = gt2/2x = vt

(If t is very small, the red

segment is nearly vertical.)

x2+ R2= (R + y)2

= R

2

+ 2

Ry + y

2

Sincey


8/19

Early AstronomersIn the 2nd century AD the Alexandrian astronomer Ptolemy put forth a

theory that Earth is stationary and at the center of the universe and that

the sun, moon, and planets revolve around it. Though incorrect, it was

accepted for centuries.

In the early 1500s the Polish astronomer Nicolaus

Copernicus boldly rejected Ptolemys geocentric model

for a heliocentric one. His theory put the sun stated that

the planets revolve around the sun in circular orbits and

that Earth rotates daily on its axis.

In the late 1500s the Danish astronomer Tycho Brahe

made better measurements of the planets and stars than

anyone before him. The telescope had yet to be

invented. He believed in a Ptolemaic-Coperican

hybrid model in which the planets revolve around the

sun, which in turn revolves around the Earth.


9/19

Early Astronomers

In the late 1500s and early 1600s the Italian scientistGalileo was one of the very few people to advocate the

Copernican view, for which the Church eventually had

him placed under house arrest. After hearing about the

invention of a spyglass in Holland, Galileo made atelescope and discovered four moons of Jupiter, craters

on the moon, and the phases of Venus.

The German astronomer Johannes Kepler was a

contemporary of Galileo and an assistant to TychoBrahe. Like Galileo, Kepler believed in the heliocentric

system of Copernicus, but using Brahes planetary data

he deduced that the planets move in ellipses rather than

circles. This is the first of three planetary laws thatKepler formulated based on Brahes data.

Both Galileo and Kepler contributed greatly to work of the English

scientist Sir Isaac Newton a generation later.


10/19

Keplers Laws ofPlanetary Motion

1. Planets move around the sun in elliptical paths with the

sun at one focus of the ellipse.

2. While orbiting, a planet sweep out equal areas in equal

times.

3. The square of a planets period (revolution time) is

proportional to the cube of its mean distance from the sun:

T2 R3

Here is a summary of Keplers 3 Laws:

These laws apply to any satellite orbiting a much

larger body.


11/19

Keplers First LawPlanets move around the sun in elliptical paths with the

sun at one focus of the ellipse.

An ellipse has two foci, F1 and F2. For any point P on the ellipse,F1P + F2P is a constant. The orbits of the planets are nearly circular

(F1 and F2 are close together), but not perfect circles. A circle is a an

ellipse with both foci at the same point--the center. Comets have very

eccentric (highly elliptical) orbits.

F1 F2

Sun

Planet

P


12/19

Keplers Second Law

Sun

While orbiting, a planet sweep out equal areas in equal times.

C

D

A

B

The blue shaded sector has the same area as the red shaded sector.

Thus, a planet moves from C to D in the same amount of time as it

moves from A to B. This means a planet must move faster when its

closer to the sun. For planets this affect is small, but for comets its

quite noticeable, since a comets orbit is has much greater eccentricity.

(proven in advanced physics)


13/19

Keplers Third LawThe square of a planets period is proportional to the

cube of its mean distance from the sun: T2 R3

Assuming that a planets orbit is circular (which is not exactly correct

but is a good approximation in most cases), then the mean distance

from the sun is a constant--the radius. F is the force of gravity on the

planet. F is also the centripetal force. If the orbit is circular, the

planets speed is constant, and v = 2R/T. Therefore,

Sun

PlanetF

RM

m

GMm

R2

mv2

R=

GMR2

m[2R/T]2

R=

Cancel ms

and simplify: 4

2

R

T 2=

Rearrange:GM

T 242

= R3

Since G, M,and are constants, T 2 R3.


14/19

Third Law Analysis

GMT 2

42= R3We just derived

It also shows that the orbital period depends on the mass of the

central body (which for a planet is its star) but not on the mass of the

orbiting body. In other words, if Mars had a companion planet the same

distance from the sun, it would have the same period as Mars,

regardless of its size.

This shows that the farther away a planet is from its star, the longer it

takes to complete an orbit. Likewise, an artificial satellite circling Earth

from a great distance has a greater period than a satellite orbiting closer.

There are two reasons for this: 1. The farther away the satellite is, thefarther it must travel to complete an orbit; 2. The farther out its orbit is,

the slower it moves, as shown:

GMm

R2

mv2

R= v =

G M

R


15/19

Third Law ExampleOne astronomical unit (AU) is the distance between Earth and the

sun (about 93 million miles). Jupiter is 5.2 AU from the sun. How

long is a Jovian year?

answer: Keplers 3rd Law says T 2 R3, so T 2= kR3, where k

is the constant of proportionality. Thus, for Earth and Jupiter we

have:

TE2= kRE

3 and TJ2= kRJ

3

ks value matters not; since both planets are orbiting the same central

body (the sun), k is the same in both equations. TE

= 1 year, and

RJ /RE = 5.2, so dividing equations:

TJ2

TE2

RJ3

RE3

= TJ2 = (5.2)3 TJ = 11.9 years



16/19

1 year

365 days

Third Law Example (cont.)What is Jupiters orbital speed?

answer: Since its orbital is approximately circular, and its speed isapproximately constant:

2 (5.2)(93106 miles)v =

dt = 11.9 years

Jupiter is 5.2 AU from the sun (5.2

times farther than Earth is).

1 day

24 hours

29,000 mph. Jupiters period from last slide

This means Jupiter is cruising through the solar system at about13,000 m/s! Even at this great speed, though, Jupiter is so far away

that when we observe it from Earth, we dont notice its motion.

Planets closer to the sun orbit even faster. Mercury, the closest

planet, is traveling at about 48,000 m/s

!


17/19

Uniform Gravitational FieldsWe live in what is essentially a uniform gravitational field. This means

that the force of gravity near the surface of the Earth is pretty much

constant in magnitude and direction. The green lines aregravitationalfield lines. They show the direction of the gravitational force on any

object in the region (straight down). In a uniform field, the lines are

parallel and evenly spaced. Near Earths surface the magnitude of the

gravitational field is 9.8 N/kg. That is, every kilogram of mass an

object has experiences 9.8 N of force. Since a Newton is a kilogram

meter per second squared, 1N/kg = 1m/s2. So, the gravitational field

strength is just the acceleration due to gravity, g.

Earths surface



18/19

Uniform Gravitational Fields (cont.)

Earths surface

10 kg

98 N

A 10 kg mass is near the surface of the Earth. Since the field

strength is 9.8 N/kg, each of the ten kilograms feels a 9.8 Nforce, for a total of 98 N. So, we can calculate the force of

gravity by multiply mass and field strength. This is the same

as calculating its weight (W = mg).


19/19

Nonuniform Gravitational FieldsNear Earths surface the gravitational field is approximately uniform.

Far from the surface it looks more like a sea urchin.

Earth

The field lines

are radial, rather than

parallel, and point toward

center of Earth.

get farther apart farther from

the surface, meaning the

field is weaker there.

get closer together closer to

the surface, meaning the

field is stronger there.

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