Gravity

1
Gravity
Newtons Law of Gravitation Keplers Laws of
Planetary Motion Gravitational Fields
2
Newtons Law of Gravitation
r
m2
m1
There is a force of gravity between any pair of
objects anywhere. The force is proportional to
each mass and inversely proportional to the
square of the distance between the two objects.
Its equation is
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
Gravity Example
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 kilometers apart?
(6.67 10-11 Nm2 / kg2) (1.23 1026 kg) (5.21
1022 kg)

(230 103 106 m) 2
8.08 1015 N
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.
4
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 2
is 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.
1.23 1026 kg
5.21 1022 kg
8.08 1015 N
8.08 1015 N
5
Inverse Square Law
The law of gravitation is called an inverse
square law because the magnitude of the force is
inversely proportional to the square of the
separation. 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
Calculating the Gravitational Constant
In 1798 Sir Henry Cavendish suspended a rod with
two small masses (red) from a thin wire. Two
larger mass (green) attract the small masses and
cause the wire to twist slightly, since each
force of attraction produces a torque in the same
direction. By varying the masses and measuring
the separations and the amount of twist,
Cavendish was the first to calculate G.
Since G is only 6.67 10-11 Nm2 / kg2, the
measurements had to be very precise.
7
Calculating the mass of the Earth
Knowing G, we can now actually calculate the mass
of the Earth. All we do is write the weight of
any object in two different ways and equate them.
Its weight is the force of gravity between it
and the Earth, which is FG in the equation
below. ME is the mass of the Earth, RE is the
radius of the Earth, and m is the mass of the
object. The objects weight can also be written
as mg.
G ME m

m g
RE 2
The ms cancel in the last equation. g can be
measured experimentally Cavendish determined
Gs value and RE can be calculated at 6.37
106 m (see next slide). ME is the only
unknown. Solving for ME we have
g RE 2
5.98 1024 kg
ME
G
8
Calculating the radius of the Earth
This is similar to the way the Greeks
approximated Earths radius over 2000 years ago
?
Red pole blocks incoming solar rays and makes a
shadow. ? is measured.
RE
s
?
The green pole is on the same meridian as the red
one, but it casts no shadow at this latitude. s
is the measured distance between the cities in
which the poles stand.
Earth
? is also the central angle of the arc. s RE
? RE s / ? ? 6.37 106 m
9
Net Force Gravity Problem
3 ? 106 kg
3 asteroids are positioned as shown, forming a
right triangle. Find the net force on the 2.5
million kg asteroid.
40 m
2.7 ? 106 kg
2.5 ? 106 kg
60 m
Steps 1. Find each force of gravity on it and
draw in the vectors.2. Find the angle at the
lower right.3. One force vector is to the left
break the other one down into
components.4. Find the resultant vector
magnitude via Pythagorean theorem
direction via inverse tangent.
answer
0.212 N at 14.6? above horizontal (N of W)
10
Falling Around the Earth
x v t
v
y 0.5 g t 2

Newton imagined a cannon ball fired
horizontally
from a mountain top at a speed
v. In a time t it
falls a distance y 0.5 g t 2 while
moving horizontally a
distance x v t. 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.

.
continued on next slide
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Necessary Launch Speed for Orbit
R Earths radiust small amount of time after
launchx horiz. distance traveled in time ty
vertical distance fallen in time t
x 2 R 2 (R y) 2 R 2 2 R y y 2 Since
y ltlt R, x 2 R 2 ? R 2 2 R y ? x 2 ? 2
R y ? v 2 t 2 ? 2 R (g t 2 / 2) ? v 2 ? R
g. So, v ? (6.37 106 m 9.8 m/s2) ½ v
? 7900 m/s
(If t is very small, the red segment is nearly
vertical.)
x v t
y g t 2 / 2
R
R
12
Early Astronomers
In 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.
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Early Astronomers
Both Galileo and Kepler contributed greatly to
work of the English scientist Sir Isaac Newton a
generation later.
In the late 1500s and early 1600s the Italian
scientist Galileo 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 a telescope 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 Tycho
Brahe. 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 that
Kepler formulated based on Brahes data.
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Keplers Laws of Planetary Motion
Here is a summary of Keplers 3 Laws
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 T 2 ? R 3
These laws apply to any satellite orbiting a much
larger body.
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Keplers First Law
Planets move around the sun in elliptical paths
with the sun at one focus of the ellipse.
F1
F2
Sun
P
Planet
An ellipse has two foci, F1 and F2. For any
point P on the ellipse, F1 P F2 P 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.
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Keplers Second Law
(proven in advanced physics)
While orbiting, a planet sweep out equal areas in
equal times.
A
D
Sun
C
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.
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Keplers Third Law
The square of a planets period is proportional
to the cube of its mean distance from the sun T
2 ? R 3
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
2 ? R / T. Therefore,
m
Cancel ms and simplify
G M
Planet
F
R 2
M
R
Sun
Rearrange
Since G, M, and ? are constants, T 2 ? R 3.
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Third Law Analysis
We 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, the farther it must travel
  to complete an orbit 2. The farther out its
  orbit is, the slower it moves, as shown

m v 2
G M

? v
R
R
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Third Law Example
One 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 ? R 3, so T 2 k R 3, where k
is the constant of proportionality. Thus, for
Earth and Jupiter we have TE 2 k RE 3 and
TJ 2 k RJ 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
TJ 2
RJ 3
? TJ 2 (5.2) 3

? TJ 11.9 years
TE 2
RE 3
continued on next slide
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Third Law Example (cont.)
What is Jupiters orbital speed? answer Since
its orbital is approximately circular, and its
speed is approximately constant
Jupiter is 5.2 AU from the sun (5.2 times
farther than Earth is).
1 year
d
2 ? (5.2) (93 106 miles)
1 day
v



t
11.9 years
365 days
24 hours
Jupiters period from last slide
? 29,000 mph.
This means Jupiter is cruising through the solar
system at about 13,000 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 !
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Third Law Practice Problem
Venus is about 0.723 AU from the sun, Mars 1.524
AU. Venus takes 224.7 days to circle the sun.
Figure out how long a Martian year is.
answer
686 days
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Uniform Gravitational Fields
We 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 are gravitational field 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,
1 N / kg 1 m/s2. So, the gravitational field
strength is just the acceleration due to gravity,
g.
continued on next slide
Earths surface
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Uniform Gravitational Fields (cont.)
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 N force, 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).
10 kg
98 N
Earths surface
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Nonuniform Gravitational Fields
Near Earths surface the gravitational field is
approximately uniform. Far from the surface it
looks more like a sea urchin.

• 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.

Earth