Download Chap. 13 Gravitational Interactions

Chapter 13
Gravitational Interactions
13.1 Gravitational Fields
A magnetic field is a force field, because
magnetic materials in it experience a force
The gravitational field around Earth is
similar
A mass in the field region experiences a
gravitational force
The field is strongest at the surface of
Earth, and declines as the distance from
Earth’s center increases
g is the acceleration due to gravity
G is the universal gravitational constant
g is the gravitational field vector
They are all different quantities that are
related
G and g have the same magnitude and
their units are equivalent: N/kg = m/s2
g = GM/R2
g = 9.8 N/kg is valid only near the planet’s
surface
as R increases, g decreases
The strength of Earth’s gravitational field is the
force per unit mass exerted by Earth on any
object
g=F
m
g = 9.8 N/kg = 9.8 m/s2
g weakens with increasing distance from Earth
Questions
If Earth had the same size but twice the
mass, what would be the acceleration of
freely falling objects at its surface?

Twice g, or nearly 20 m/s2
If Earth has the same mass but half the
size, what would be the acceleration of
freely falling objects at its surface?

Four times g, or nearly 40 m/s2
If Earth had twice the mass, and half the
size, what would be the acceleration of
freely falling objects at its surface?

Eight times g, or nearly 80 m/s2
What evidence would you look for to tell
whether or not you were in a gravitational
field?

The presence of a gravitational force
Compared to its strength at Earth’s
surface, what is the strength of the
gravitational field at a distance of two
Earth radii from the center of Earth?

¼ as much
13.2 Gravitational Field Inside a
Planet
Gravitational field on Earth exists inside
Earth as well as outside
Imagine a hole drilled completely through
Earth
FALLING THROUGH THE
CENTER OF THE EARTH
Start at the North Pole
Fall & gain speed down
to the center
Lose speed all the way to
the South Pole
Gain speed toward the
center
Lose speed moving away
from the center
a=g
a=g/2
a=0
a=g/2
a=g
As you fell faster and faster into the Earth,
your acceleration diminishes because the
pull of the mass above you partly cancels
the pull below
At the Earth’s center the pulls cancel to
zero and your acceleration is zero
Momentum carries you against a growing
acceleration past the center to the opposite
side where it is again, g
STUCK IN A CAVITY
In a cavity at the center of the Earth, your
weight would be zero, because you would
be pulled equally by gravity in all directions
You are pulled in every direction equally
The gravitational field at Earth’s center is
zero
Questions
If you dropped a rock into a tunnel through
Earth, what would happen?

It would gain speed until it reached Earth’s
center, and then lose speed the rest of the
way. Its speed at the far end of the tunnel
would be the same as its initial speed. It
would then fall back and repeat the motion in
cyclic fashion.
As the velocity of the rock increases when
falling into the Earth tunnel, what happens
to the acceleration?

It decreases as the gravitational field
decreases, and is zero at Earth’s center. The
falling body has its maximum velocity at
Earth’s center, where both the field and
acceleration are zero
13.3 Weight and Weightlessness
Weight: support force

We are as heavy as we feel
Weightlessness: the absence of a support
force

The queasy feeling you experience when in a
car that speeds over the top of a hill
FOR EXAMPLE: A SCALE
Stand on a bathroom scale on the floor
The gravitational force between you and
Earth pulls you against the supporting floor
and scale
By Newton’s 3rd law, the floor and scale in
turn push upward on you
NOW…WEIGHT IN AN
ELEVATOR
UP, UP, AND AWAY
If the elevator
accelerates
upward, the
bathroom scale
and floor would
push harder
against your feet
The scale would
show an increase
in your weight
GROUND FLOOR PLEASE
If the elevator
accelerated
downward, the
scale would show
a decrease in your
weight
The support force
of the floor would
now be less
ELEVATOR CABLE BROKE
Elevator falls freely,
the scale would read
zero
According to the
scale, you would be
weightless and feel
weightless
Your insides would no
longer be supported
by your legs and hips
Question
Why would you feel weightless in an
elevator with a broken cable?

There would be no support force-the floor
would fall as fast as you
13.4 Ocean Tides
Think of someone pulling on your coat
If some one pulled only on the sleeve, the
coat could tear
But if every part of your coat were pulled
equally, it and you would accelerate, but
the coat wouldn’t tear
It tears when one part is pulled harder
than another because of a difference in
forces acting on the coat
In a similar way, the spherical Earth is
“torn” into an elliptical shape by
differences in gravitational forces exerted
by the moon
Ocean tides are caused by differences in
the gravitational pull of the moon on
opposite sides of Earth
The moon’s attraction is stronger on
Earth’s oceans closer to the moon, and
weaker on the oceans farther from the
moon

gravitational forces weakened with increased
distance
Why are there two tides a day?
There would only be one tide per day, if the Earth
were “nailed down” in one place and held stationary,
except for a daily rotation
But the Earth and moon are in orbit around each
other
The ocean nearest the moon is pulled upward
toward the moon, while the main body of Earth is
pulled toward the moon

away from the ocean on the far side
Earth is closer to the moon than the far-side ocean
is, so Earth’s waters get slightly elongated-at both
ends creating ocean bulges
Earth makes one complete turn per day beneath
these ocean bulges creating two sets of ocean
tides per day
The part of Earth that passes beneath one of the
bulges has a high tide
The tide cycle is every 24 hours and 50 minutes
The sun also contributes to ocean tides, but not
as much as the moon.
The tilt of Earth’s axis is another factor
Questions
Which pulls harder on the oceans of Earth,
the sun or the moon?

The sun
Which is most effective in raising tides?

The moon
The difference in pulls decreases as the
cube of the distance between the centers of
the bodies


Twice as far away produces 1/8 the tide
3 times as far, only 1/27 the tide
Only relatively close distances result in
appreciable tides, and so the nearby moon
“out-tides” the enormously more massive
but far away sun
When the sun, Earth, and the moon are
lined up, the tides due to the sun and the
moon coincide
There are higher high tides and lower low tides
These are called spring tides

They have nothing to do with spring
If the alignment is perfect, there is an
eclipse
ECLIPSE
A lunar eclipse is produced when Earth is
directly between the sun and moon
A solar eclipse is produced when the
moon is directly between the sun and
Earth
PHASES OF THE MOON
When the Earth is between the sun and the
moon –full moon
When the moon is between the sun and Earth –
new moon
Spring tides occur during a new moon and full
moon
During half-moons, the tides due to the sun and
the moon partly cancel each other
 High tides are lower than average
 Low tides are higher than average
These are called neap tides
Question
At the time of extra high tides, will extra
low tides follow in the same day?

Yes, by the “conservation of water”. There is
only so much water on Earth-extra high tides
in one part of the world means extra low tides
in another
13.5 Tides in Earth and the
Atmosphere
There are tides within Earth, which is
mostly molten lava
There is a greater probability of
earthquakes and volcanoes when there is
an eclipse of the sun or the moon
This is when Earth experiences spring
tides-greater stresses on Earth’s crust
If the moon were closer…
Ocean tides would be higher and so the
tidal forces on the moon’s crust would be
greater
If the moon were too close, Earth’s tidal
forces would tear the moon into a billion
pieces, forming a ring around Earth similar
to those around Saturn
13.6 BLACK HOLES
Occurs with stars that are at least two
to three times more massive than our
sun
Once the flame of thermonuclear
fusion is extinguished, gravitational
collapse takes over-and it doesn’t stop
Gravitation is so enormous that
nothing can get back out-even light
A collapsed star represents condensed mass
and therefore condensed gravity.
The mass of a black hole is no more than the
mass of the star that collapsed to form it
Hence the gravitational field of the star and the
black hole are the same at distances greater
than the original radius of the star
It is only at closer distances that the enormous
field occurs
Question
Consider a satellite companion to a star
that collapses to become a black hole.
How will the orbit of the companion
satellite be affected by the star’s
transformation to a black hole?

Not at all. No terms in the gravitational
equation change
Summary
Earth can be thought of as being surrounded by
a gravitational field that interacts with objects and
causes them to experience gravitational forces.
The gravitational field, g, is equal to the
acceleration of a freely falling object.
Objects in orbit around Earth have a gravitational
force acting on them even though they may
appear to be weightless.
Ocean tides (and even tides within the solid
Earth and within the atmosphere) are
caused by differences in the gravitational
pull of the moon (and sun) on opposite
sides of Earth.
When a star runs out of fuel for fusion, it
collapses under gravitational forces.
Sufficiently massive stars collapse to form
black holes.