The Universal Force of Gravity

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Module 4 The Wanderers
Activity 2 The Universal Force of Gravity
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Summary
In this Activity, we will investigate (a)
elliptical orbits and Keplers Laws, (b) Newtons
Law of Gravitation, and (c) apparent
weightlessness in orbit.
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(a) Elliptical Orbits and Keplers Laws

• Some orbits in the Solar System cannot be
  approximated at all well by circles

- for example, Plutos separation from the Sun
varies by about 50 during its orbit!
According to Keplers First Law, closed orbits
arounda central object under gravity are
ellipses.
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As a planet moves in an elliptical orbit, the Sun
is at one focus (F or F) of the ellipse.
v
r
C
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The line that connects the planets point of
closest approachto the Sun, the perihelion ...
v
perihelion
r
C
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and its point of greatest separation from the
Sun, the aphelion
As a planet moves in an elliptical orbit, the Sun
is at one focus (F or F) of the ellipse
v
perihelion
is called the major axis of the ellipse.
r
C
aphelion
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The only other thing we need to know about
ellipses is howto identify the length of the
semi-major axis, because that determines the
period of the orbit.
Semi means half, and so the semi-major axis a
is half thelength of the major axis
v
r
C
F
F
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For circular orbits around one particular mass -
e.g. the Sun - we saw that the period of the
orbit (the time for one completerevolution)
depended only on the radius r
- that was Keplers 3rd Law
M
For objects orbiting a common central body (e.g.
the Sun)in approximately circular orbits,
r
r
m
v
the orbital period squared is proportional to
the orbital radius cubed.
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Now heres the mathematics.
where G and 4p2 are constants and M is the mass
of the Sun
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Lets see what determines the period for an
elliptical orbit
For elliptical orbits,the period dependsnot on
r, but on thesemi-major axisa instead.
v
r
C
F
F
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It turns out that Keplers 3rd Law applies to
all ellipticalorbits, not just circles, if we
replace orbital radiusby semi-major axis
For objects orbiting a common central body (e.g.
the Sun)
the orbital period squared is proportional to
the orbital radius cubed.
the orbital period squared is proportional to
the semi-major axis cubed.
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Each of these orbits has the same semi-major
axis length a
Note that a circleis a special case ofan
ellipse, wherer a.
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So if each of these orbits is around the same
massiveobject (e.g. the Sun),
then as they all have the same semi-major axis
length a,
then, by KeplersThird Law, they have the
sameorbital period.

• Click here to see a simulation illustrating
  Keplers Third Law.

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• So, as you saw in the simulation, bodies orbiting
  at large distances have much longer orbital
  periods.

For the mathematicallyinclined, the square of
the period P of the orbit increases in
proportion to the cube of the semi-majoraxis
a
distant planets havemuch large orbital periods
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We havent yet met Keplers Second Law.
Thats because its not at all interesting for
circular, oralmost circular orbits.
But if we look at a quite eccentric elliptical
orbit, for example, that of Halleys comet
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Comet Halley in 1910
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Neptune
Sun
Note that Comet Halleys orbit is retrograde,
which means that it orbits the Sun clockwise when
viewed from the north pole. This is the the
opposite sense to that of the planets.
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An object in a highly elliptical orbit travels
very slowlywhen it is far out in the Solar
System,
but speeds up as it passes the Sun.
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According to Keplers Second Law,
the line joining the object and the Sun ...
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sweeps out equal areas in equal intervals of
time.
equal areas
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That is, Keplers Second Law states that
the line joining a planet and the Sun sweeps
outequal areas in equal intervals of time.
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(b) Newtons Law of Gravitation

• We call the force which keeps the Moon in its
  orbit around the Earth gravity.

Sir Isaac Newtons conceptual leap in
understandingof the effects of gravity largely
involved his realisationthat the same force
governs the motion of a falling objecton Earth -
for example, an apple - and the motion of the
Moon in its orbit around the Earth.
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• Isaac Newton discovered that two bodies share a
  gravitational attraction, where the force of
  attraction depends on both their masses

MSun
MEarth
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• Both bodies feel the same force, but in opposite
  directions.

MSun
MEarth
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This is worth thinking about - for example, drop
a pen to the floor. Newtons laws say that the
force with which the pen is attracting the Earth
is equal and opposite to the force with which
the Earth is attracting the pen, even though the
pen is much lighter than the Earth!
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• Newton also worked out that if you keep the
  masses of the two bodies constant, the force of
  gravitational attraction depends on the distance
  between their centres

mutual force of attraction
Note that the gravitational force is larger the
closer the objects are together.
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• For any two particular masses, the gravitational
  force between them depends on their separation
  as

as the separation between the masses is
increased, the gravitational force of
attractionsbetween them decreases quickly.
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• Your pen dropping to the floor and a satellite in
  orbit around the Earth have something in common -
  they are both in freefall.

To see this, remember Newtons thought experiment
from the Activity Solar System Orbits
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On all these trajectories, the projectile is in
free fall under gravity. (If it were not, it
would travel in a straight line - thats
NewtonsFirst Law of Motion.)
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If the ball is not given enough sideways
velocity, its trajectory intercepts the Earth
- that is, it falls to Earth eventually.
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On the trajectories which make complete orbits,
the projectile is travelling sideways fast
enough ...
On all these trajectories, the projectile is in
free fall.
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that as it falls, the Earth curves away
underneathit, and the projectile completes
entire orbits without ever hitting the Earth.
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(c) Apparent Weightlessness in Orbit
This astronaut on a space walk is alsoin free
fall.
The astronauts sideways velocityis
sufficient to keephim or her in orbitaround the
Earth.
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Lets take a little time to answer the following
question

• Why do astronauts in the Space Shuttle in Earth
  orbit feel weightless?

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• Some common misconceptions which become apparent
  in answers to this question are

(a) there is no gravity in space, (b) there is no
gravity outside the Earths atmosphere, or (c) at
the Shuttles altitude, the force of gravity is
very small.
Click on each alternative to see why we claim
that its a misconception!
Then see if you agree with our explanation ...
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In a spacecraft (like the Shuttle) in Earth
orbit, astronauts are in free fall, at the same
rate as their spaceships.
On all these trajectories, the projectile is in
free fall.
That is why they experience weightlessness just
as a platform diver feels while diving down
towards a pool, or a sky diver feels while in
free fall.
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In the next Module well look at one Solar
Systemorbit in particular - that of the Moon
around the Earth.
On all these trajectories, the projectile is in
free fall.
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Image Credits
NASA View of Australia http//nssdc.gsfc.nasa.gov
/image/planetary/earth/gal_australia.jpg NASA
Halley in 1910 http//pds.jpl.nasa.gov/planets/gif
/smb/hal1910.gif NASA Space Shuttlehttp//lisar.
larc.nasa.gov/LISAR/IMAGES/SMALL/EL-1994-00718.jpe
g
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• Now return to the Module home page, and read more
  about gravity in the Textbook Readings.

Hit the Esc key (escape) to return to the Module
4 Home Page
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(a) There is no gravity in space?

• At face value, this statement doesnt bear too
  much examination, because Newtons Law of
  Gravitation has been applied right from its
  inception to the motion of the Moon and planets -
  and they are in space.

When people make this assumption, perhaps what
theyare really saying is that the sort of
gravity which makesus feel heavy only exists on
planetary surfaces - but Newton developed the Law
in the first place by realizing that gravity as
it acts on Earth (e.g. on an apple)is the same
force as that which acts on the Moon and planets.
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Back to the alternative answers
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(b) There is no gravity outside the Earths
atmosphere?

• Like (a), at first glance this misconceptions
  seems naïve, because Newtons Law of Gravitation
  has been applied right from its inception to the
  motion of the Moon and planets - and they are in
  space.

However what this statement might really be
revealing is a link many people perceive between
gravity and air in other words, the mistaken
idea that gravity does not exist in a vacuum -
that air, in some way, makes things heavy.
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• In fact, air actually makes objects feel very,
  very slightly lighter - like buoyancy in a tank
  full of water, except that air is so much less
  dense than water that the effect is not
  noticeable.

The misconception that links gravity and air
shows up in some science fiction movies too -
watch for the one where a Concorde-type plane
mistakenly ends up in Earth orbit. The
passengersinside the plane can walk around, with
a bit of care, butastronauts sent up to help
them float weightlessly aboutoutside!
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Back to the alternative answers
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(c) At the Shuttles altitude, the force of
gravity is very small?

• This statement sounds reasonable - after all, the
  Shuttle is way out in space - until you check it
  with calculations.

In fact, compared to the radius of the Earth
(6378 km), atypical Shuttle altitude above the
Earths surface of 200 kmor so is pretty
negligible. At that altitude, the force of
gravity is only 5 less than on the Earths
surface.
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Back to the alternative answers
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