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Gravity
Gravity is the weakest of the four fundamental forces, yet it is the dominant force in the
universe for shaping the large scale structure of galaxies, stars, etc. The gravitational
force between two masses m1 and m2 is given by the relationship:
This is often called the "universal law of gravitation" and G the universal gravitation
constant. It is an example of an inverse square law force. The force is always attractive
and acts along the line joining the centers of mass of the two masses. The forces on the
two masses are equal in size but opposite in direction, obeying Newton's third law.
Viewed as an exchange force, the massless exchange particle is called the graviton.
The gravity force has the same form as Coulomb's law for the forces between electric
charges, i.e., it is an inverse square law force which depends upon the product of the two
interacting sources. This led Einstein to start with the electromagnetic force and gravity
as the first attempt to demonstrate the unification of the fundamental forces. It turns out
that this was the wrong place to start, and that gravity will be the last of the forces to
unify with the other three forces. Electroweak unification (unification of the
electromagnetic and weak forces) was demonstrated in 1983, a result which could not be
anticipated in the time of Einstein's search. It now appears that the common form of the
gravity and electromagnetic forces arises from the fact that each of them involves an
exchange particle of zero mass, not because of an inherent symmetry which would make
them easy to unify.
Fundamental Forces
Weight
The weight of an object is defined as the force of gravity on the object and may be
calculated as the mass times the acceleration of gravity, w = mg. Since the weight is a
force, its SI unit is the newton.
For an object in free fall, so that gravity is the only force acting on it, then the expression
for weight follows from Newton's second law.
You might well ask, as many do, "Why do you multiply the mass times the freefall
acceleration of gravity when the mass is sitting at rest on the table?". The value of g
allows you to determine the net gravity force if it were in freefall, and that net gravity
force is the weight. Another approach is to consider "g" to be the measure of the intensity
of the gravity field in Newtons/kg at your location. You can view the weight as a measure
of the mass in kg times the intensity of the gravity field, 9.8 Newtons/kg under standard
conditions.
Data can be entered into any of the boxes below. Then click outside the box to update the
other quantities.
At the Earth's surface, where g=9.8 m/s2 :
The weight of mass
kg is
The weight of mass
slugs is
Newtons
pounds
The kilogram is the SI unit of mass and it the
almost universally used standard mass unit.
The associated SI unit of force and weight is
the Newton, with 1 kilogram weighing 9.8
Newtons under standard conditions on the
Earth's surface. However, in the US common
units, the pound is the unit of force (and
therefore weight).The pound is the widely
used unit for commerce. The use of the pound
force constrains the mass unit to an
inconveniently large measuring unit called a
"slug". The use of this unit is discouraged,
and the use of exclusively SI units for all
scientific work is strongly encouraged.
What is a slug?
What is a Slug?
The slug is the unit of mass in the US common system of units, where the pound is the
unit of force. The pound is therefore the unit of weight since weight is defined as the
force of gravity on an object. While the pound force and pound weight are the widely
used units for commerce in the United States, their use is strongly discouraged in
scientific work. The standard units for most of scientific work are the SI units.
Mass and Weight
The mass of an object is a fundamental property of the object; a numerical measure of its
inertia; a fundamental measure of the amount of matter in the object. Definitions of mass
often seem circular because it is such a fundamental quantity that it is hard to define in
terms of something else. All mechanical quantities can be defined in terms of mass,
length, and time. The usual symbol for mass is m and its SI unit is the kilogram. While
the mass is normally considered to be an unchanging property of an object, at speeds
approaching the speed of light one must consider the increase in the relativistic mass.
The weight of an object is the force of gravity on the object and may be defined as the
mass times the acceleration of gravity, w = mg. Since the weight is a force, its SI unit is
the newton. Density is mass/volume.
Earth's Gravity
The weight of an object is given by W=mg, the force of gravity, which comes from the
law of gravity at the surface of the Earth in the inverse square law form:
At standard sea level, the acceleration of gravity has the value g = 9.8 m/s2, but that value
diminishes according to the inverse square law at greater distances from the earth. The
value of g at any given height, say the height of an orbit, can be calculated from the
above expression.
Above the earth's surface at a height of h =
corresponds to a radius r =
m/s2 =
m=
x 106 m, which
x earth radius, the acceleration of gravity is g =
x g on the earth's surface.
Please note that the above calculation gives the correct value for the acceleration of
gravity only for positive values of h, i.e., for points outside the Earth. If you drilled a hole
through the center of the Earth, the acceleration of gravity would decrease with the radius
on the way to the center of the Earth. If the Earth were of uniform density (which it is
not!), the acceleration of gravity would decrease linearly to half the surface value of g at
half the radius of the Earth and approach zero as you approached the center of the Earth.
Hole through center of Earth
Journey through the center of the Earth
Suppose you could drill a hole through the Earth and then
drop into it. How long would it take you to pop up on the
other side of the Earth?
Your initial acceleration would be the surface acceleration of gravity
but the acceleration would be progressively smaller as you approached the center. Your
weight would be zero as you flew through the center of the Earth. For our hypothetical
journey we will assume the Earth to be of uniform density and neglect air friction and the
high temperature of this trip.
For a spherically symmetric mass, the net gravity
force on an object from that mass would be only
that due to the mass inside its radius, and that
would act as if it were a point mass located at the
center. When this is analyzed in detail, you find
that the gravity at any radius r less than REarth will
be linearly proportional to the distance from the
center.
Gravity force of spherical shell
On mass outside the shell
On mass inside the shell
Taking positive r as outward from the center of the Earth:
This is the same form as Hooke's Law for a mass on a spring. It would cause the transEarth traveler to oscillate back and forth through the center of the Earth like a mass
bobbing up and down on a spring. The angular frequency and period for this oscillation
are
For this case the period of oscillation is
The traveler accelerates toward the center of the Earth and is momentarily weightless
when passing through the geometric center at about 7900 m/s or almost 17,700 miles/hr.
The traveler would pop up on the opposite side of the Earth after a little more than 42
minutes. But unless he or she grabs something to hold on, they will fall back for a return
journey and continue to oscillate with a round-trip time of 84.5 minutes.
As a further feature of this fanciful journey,
suppose a satellite could be put in a circular
orbit about the Earth right above the surface.
Ignoring air drag and the terrific sonic boom that
would accompany such an orbit, suppose it
passed overhead just above the falling person as
they popped up out of the hole. The period of
such an orbit would be such that it would be
passing overhead every time the oscillating
person popped up on either side of the Earth.
The period of the orbit is calculated from
which is the same as the period of the oscillating body.
Other examples of weightlessness
Weightlessness
While the actual weight of a person is determined by his mass and the acceleration of
gravity, one's "perceived weight" or "effective weight" comes from the fact that he is
supported by floor, chair, etc. If all support is removed suddenly and the person begins to
fall freely, he feels suddenly "weightless" - so weightlessness refers to a state of being in
free fall in which there is no perceived support. The state of weightlessness can be
achieved in several ways, all of which involve significant physical principles.
Click on any of the examples for further details.
Inverse Square Law, Gravity
As one of the fields which obey the general inverse square law, the gravity field can be
put in the form shown below, showing that the acceleration of gravity, g, is an expression
of the intensity of the gravity field.
Tides
The Earth experiences two high tides per day because of the difference in the Moon's
gravitational field at the Earth's surface and at its center. You could say that there is a
high tide on the side nearest the Moon because the Moon pulls the water away from the
Earth, and a high tide on the opposite side because the Moon pulls the Earth away from
the water on the far side. The tidal effects are greatly exaggerated in the sketches.
Why is the moon's tidal effect greater than the Sun's?
How large is the Sun's tidal effect compared to the moon's influence?
Moon as Dominant Tidal Source
The Moon is the dominant tidal influence
because the fractional difference in its
force across the Earth is greater than the
fractional difference seen from the Sun.
This difference in force follows the
inverse square law.
Sun's tidal influence
Sun's Tidal Effect
Even though the Sun is 391 times as far away from the Earth as the Moon, its force on the
Earth is about 175 times as large. Yet its tidal effect is smaller than that of the Moon
because tides are caused by the difference in gravity field across the Earth. The Earth's
diameter is such a small fraction of the Sun-Earth distance that the gravity field changes
by only a factor of 1.00017 across the Earth. The actual force differential across the Earth
is 0.00017 x 174.5 = 0.03 times the Moon's force, compared to 0.068 difference across
the Earth for the Moon's force. The actual tidal influence then is then 44% of that of the
Moon.
Why is the moon the dominant tidal influence?
Moon as Dominant Tidal Source
The Moon is the dominant tidal influence
because the fractional difference in its
force across the Earth is greater than the
fractional difference seen from the Sun.
This difference in force follows the
inverse square law.
Sun's tidal influence
Jupiter Effect
A popular book warned of disastrous gravitational effects from the special alignment of
the planets Jupiter and Mars along with the Moon. The forces of attraction compared to
that of the Earth are illustrated for a person weighing 160 lb on the Earth's surface.
Don't lose much sleep worrying about the Jupiter effect. You change the gravity force on
yourself by taking one step up a stairway more than the combined gravitational effects of
both Jupiter and Mars if they were perfectly aligned!