Download Chapter 7 Gravitation

Objectives: The student will be able to:
1.Apply the proportional relationship of
the law of universal gravitation.
2.Use Newton’s second law and the law
of universal gravitation to show why
objects near the surface of the earth
fall with the same constant
acceleration.
3.Explain why a spaceship in a stable
circular orbit is in free fall and why
a person in that spaceship experiences
weightlessness.
The Big Idea
• Everything pulls
everything else.
• There is a force that pulls all
objects together. It is gravity.
What Newton
Knew
Newton understood
the concept of inertia
developed earlier by Galileo.
•Without an outside force, moving objects continue
to move at constant speed in a straight line.
•If an object undergoes a change in speed or
direction,
then a force is responsible.
Newton’s 1st Law
The Law of Inertia
• What is it?
– An object in equilibrium will remain in
equilibrium unless acted on by a non
zero net force.
• Equilibrium
– Zero Net Force.
– No Acceleration.
• Static - Object at rest.
• Dynamic - Object moving
at a constant speed in a straight line.
The Apple and the Moon
• Newton saw apples falling to
Earth and wondered if the
moon fell towards the Earth
just like the apple fell towards
the Earth.
•Was he correct?
•What makes things fall towards the
center of the Earth?
•What is different about the moon and
the apple?
The Moon Falls?
• If something is moving and
no force acts on it, how does it
Keep moving?
• What is needed for circular
motion?
Newton realized that if the moon did not fall, it
would move off in a straight line and leave its
orbit.
His idea was that the moon must be falling around
the Earth.
Cannonball being shot off
a Very Tall Mountain
Nevvton’s
Thought
aExperiment.
n
• If I could climb a mountain tall enough, could
I shoot a cannonball so that it would never
land back on Earth.
• We call this putting an object into orbit.
I need to
test my
hypothes
is
From hypothesis
to theory
• Newton thought the apple, the orbiting
cannonball, and the motion of the moon
were all caused by a force now called
gravity.
• He needed to test this hypothesis.
The moon and the apple.
• Newton knew the apple fell 5m in one
second.
• He wondered how far the moon fell in one
second.
• The moon was 60 times away from the
Earth than the apple was.
• The force of Gravity must dilute the farther
away something is.
Using geometry, Newton calculated how far the circle of
the moon’s orbit lies below the straight-line distance
the moon otherwise would travel in one second. His
value turned out to be about the 1.4-mm distance
accepted today.
But he was unsure of the exact Earth moon distance, and
whether or not the correct distance to use was the
distance between their centers. At this time he hadn’t
proved mathematically that the gravity of the spherical
Earth (and moon) is the same as if all its mass were
concentrated at its center.
Because of this uncertainty, and also because of criticisms
he had experienced in publishing earlier findings in optics,
he placed his papers in a drawer, where they remained
for nearly 20 years.
During this period he laid the foundation and developed the
field of geometrical optics for which he first became
famous.
Newton finally returned to the moon problem at the prodding
of his astronomer friend Edmund Halley (of Halley’s
comet fame). It wasn’t until after Newton invented a new
branch of mathematics, calculus, to prove his center-ofgravity hypothesis, that he published what is one of the
greatest achievements of the humankind, the law of
universal gravitation.
Very fast horizontal toss
t = 0s
V=8km/s
t = 1s
x= 8km
5m
t = 2s
x=16km
20m
t = 3s
x=24km
45m
Orbital motion is free fall
(stopped here)
Circular Orbit!
Eliptical
Orbit
V = 10
4 km/s
6
8
km/s
Centripetal acceleration
a = v2/r for a circular orbit
(v = 8km/s = 8x103m/s)
(8 x103 m/s)2
64 x106 m2/s2
a=
=
6
6.4 x 10 m
6.4 x 106 m
= 10 m/s2
Toward Earth’s center
=g
Artificial satellite
a
v
a = v2/r
= g
Moon-earth
v
a=v2/r
Is the Moon in free-fall around the
Earth?
a = v2/r
what is v?
2pr
v = dist/time =
28d
24 x 108 m
=
6
x
2.4 10 s
2x3.14x 3.84x108 m
= 28dx(24h/d)x3.6x103s
= 1.0 x 103 m/s
Moon’s centripetal acceleration
amoon = v2/r;
v = 1.0 x103 m/s)
(1.0 x103 m/s)2 1.0x106 m2/s2
amoon=
=
8
3.84 x 10 m
3.84 x 108 m
= 2.7
x
10-3
m/s2
Toward Earth’s center

1
3600
g
Newton’s dreams
Hmmmmm
The Moon is in free-fall
around the Earth
It’s acceleration is only
1/3600 g (accel at the
Earth’s surface)
Distances
The moon is 60x further from
the Earth’s center than objects
on (near) the Earth’s surface
r=3.84x108m = 60 x 6.4x106 m
= 60 x RE
RE = 6.4x106m
(
1 2
1
60 ) = 3600
Newton’s
big idea
The moon is 60x
further from
the Earth’s center
than objects on
(near) the Earth’s
surface
The strength of
Earth’s gravity
near the Moon is
(1/60)2 t=1/3600
times weaker
Gravity gets weaker as 1/dist2
inverse-square law
Universal law of gravity
m
r
M
F  m
F  M
combine:
1
F 
r2
mM
F 
r2
Proportionality constant:
“Newton’s Constant”
mM
F = G r2
Universal
Universal:
applies to all objects!!!
What is G?
W
mME
W= G R 2
E
GME
W= m R 2
E
W= m g
GME
g= R 2
E
Force of gravity between
“ordinary-sized” objects
80kg
mM
F = G r2
1m
60kg
F = 6.7x10-11Nm2/kg2
60 kg 80kg
(1m)2
F = 6.7x60x80x10-11N
F =
32160.x10-11 N
= 3.2x10-7 N
 30x109 times bigger!
Boy’s weight = mg = 80kg x 10m/s2 = 800 N
Measuring G
• G was first measured 150 years after
Newton’s discovery of universal
gravitation by an English physicist, Henry
Cavendish.
Henry
Cavendish’
s
experiment
determined the
proportionality
constant
G
in 1798.
Detailed clip on experiment
http://www.newscientist.com/data/images/archive/1639/16390101.jpg
Newton’s Law of Universal Gravitation
The gravitational force on you is one-half of a
Third Law pair: the Earth exerts a downward force
on you, and you exert an upward force on the
Earth.
When there is such a disparity in masses, the
reaction force is undetectable, but for bodies
more equal in mass it can be significant.
Measuring Weight
N
mg
Weightlessness
N =0
N =mg
N >mg
N <mg
Weightlessness means N =0
compensating upward
Weightlessness in action
“Floating” is space is
really free-falling in space
What is g on
the moon?
mMM
W= G R 2
M
W
MM
GMM
W= m R 2
M
W= m gM
GMM
g M= R 2
M
gM on the Moon
GMM
g = R 2
M
=
6.7x10-11Nm2/kg2 x 7.4x1022Kg
(1.7x106m)2
gM = 1.7 m/s2
1/6 x
gEarth
•How does the acceleration of gravity depend on the mass
of a falling object?
•It does not. All falling objects fall with the same
acceleration (on a particular planet).
•Now see why…
•F = ma and on Earth acceleration due to gravity
denoted “g” so F=mg or g=F/m
•If mass of earth is M1 then Fg=GM2/d2
© 2004 Pearson Education Inc., publishing as Addison-Wesley
Thus, the weight of an
object of mass m at the
surface of the Earth is
obtained by multiplying
the mass m by the
acceleration due to
gravity, g, at the surface of
the Earth. The
acceleration due to gravity
is approximately the
product of the universal
gravitational constant G
and the mass of the Earth
M, divided by the radius
of the Earth, r, squared.
Gravity Near the Earth’s Surface;
Geophysical Applications
The acceleration due to
gravity varies over the
Earth’s surface due to
altitude, local geology,
and the shape of the
Earth, which is not quite
spherical.
Satellites and “Weightlessness”
Satellites are routinely put into orbit around the
Earth. The tangential speed must be high
enough so that the satellite does not return to
Earth, but not so high that it escapes Earth’s
gravity altogether.
Satellites and “Weightlessness”
The satellite is kept in orbit by its speed – it is
continually falling, but the Earth curves from
underneath it.
Satellites and “Weightlessness”
Objects in orbit are said to experience
weightlessness. They do have a gravitational
force acting on them, though!
The satellite and all its contents are in free fall, so
there is no normal force. This is what leads to the
experience of weightlessness.
Satellites and “Weightlessness”
More properly, this effect is called apparent
weightlessness, because the gravitational force
still exists. It can be experienced on Earth as
well, but only briefly:
• The velocity of a satellite keeps it in orbit
• Even when moving, the satellite is actually
accelerating toward the Earth (this is what
keeps it in its circular path)
• Its acceleration results in a curved path which
is the same as the curve of the Earth
• Gravity is providing the centripetal force
Perception of
Weightlessness
• There is still gravity acting in a satellite
(about 8.9 m/s2), so why do we feel
weightless?
• In an free falling elevator, if the FA is equal
to the FG, there is no FN
• No force is felt feel weightless – called
apparent weightlessness
• Weightlessness that you feel in a
satellite is like the weightlessness in an
elevator
• The satellite and everything on it are all
accelerating toward the earth at the
same rate
Video Period 3
• The Apple and the Moon
Problem 1
•
Two spheres of mass 35kg are 30m apart.
A) What force does one exert on the other?
B) If the mass of one is tripled and the radius
is quadrupled how does the force change?
Problem 2
• Two spheres of equal mass have a force
of gravity of 7x10-9 N exerted on each
other. If the distance between them is 7m,
find the mass.
Problem 3
• Find the value of the gravitational
acceleration g. The mass of the Earth is
6.0 x 1024kg. The radius of the Earth is
6.38 x 106 m.
Elaboration
• Gravitation worksheet
• Law of Universal Gravitation.
Homework – Chapter 7
• Kahoot