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(This article originally appeared in the Q1 1993 STAR news letter. -Ian)
The
Basics of Gravitational
Force
and Motion
by
Jerry Watson
In
the latter 1660's when Isaac Newton was about 25 years old, much of
his innovative work concerning the relationship of force and motion
had already been formulated. Of particular interest here are his
three laws of motion and the law of universal gravitation. These
insights into the physical workings of the natural world, and the new
mathematics needed to quantify them (calculus), were not formally
introduced to the world until 1687 in Newton's book Principia.
This
subject matter is usually found in the early chapters of any basic
textbook on astronomy. I can recommend Introductory
Astronomy and Astrophysics
by Michael Zeilick and Elske Smith, 2nd edition (1987), CBS College
Publishing, pp. 503. Chapter 1, entitled "Celestial Mechanics
and the Solar System" is written at the undergraduate college
level. The discussion to follow is a considerably condensed summary
of this material with the purpose of getting at the basics. This
level of understanding will be adequate to explore the amazing
acrobatics of celestial objects moving under the influence of their
gravitational attraction.
1.
The Force of Gravity
We
shall consider the simplest of all physical systems: just two
objects alone in the universe. Much can be learned from this
so-called 'two-body problem', even about the behavior of planets,
satellites, and other solar system objects under the control of the
sun. For such a system Newton proposed that the force (F) of object
1 on object 2, and vice versa, is given by:
On
the right hand side m1
and m2
represent the respective masses
of the two objects. Sometimes in our everyday conversation "mass"
and "weight" are used synonymously; they are related, but
they are not the same. Newton meant by "mass" the quantity
of matter (atoms and molecules) of which something is composed. The
Apollo astronauts took their mass with them when they went to the
Moon. However, on the Moon their mass weighed only about one-sixth
the value at the Earth's surface. As a matter of fact your "weight"
is actually a measure of the 'force of gravity' exerted on you by the
mass of the other object (e.g., the Earth or Moon).
A
basic unit of mass, the gram,
was defined as the mass of 1 cubic centimeter of pure water. The
usual unit of mass for calculation purposes is the kilogram
(kg), equal to 1000 grams. At the Earth's surface (strictly, at mean
sea level) 1 kg weighs 2.2 pounds. Hence, a 220 pound object has a
mass of 100 kg. On the Moon, that mass would weigh about 220/6 ≈
37 pounds.
The
law of gravitation in (1) above tells us that the force is
proportional to the product of the two masses, and is inversely
proportional to the square
of the distance (r) between the two objects. Newton proved that even
when dealing with large spherical objects (e.g., Sun or Earth), the
objects gravitationally interact with one another as though all of
their mass were concentrated at their respective centers. Thus, r is
the distance between the centers of the two objects. The fact that
the force depends inversely on the 'square' of the separation means
that the force decreases rapidly with increasing distance (and vice
versa). If the distance between two masses is doubled, the
attractive force decreases to one-fourth; if the separation becomes
10 times some original value, then the force is only 1/100th of the
original. Relationships of the form of equation (1) are referred to
as 'inverse square' laws.
Finally,
the right hand sides of equation (1) contain the universal constant
of gravitation (G). The determination of the value of this constant
has been a challenge for experimentalists since the first attempt by
Henry Cavendish in England in 1798. A summary of the history of
measurement apparatuses, and also of the reasons why an accurate
value of G is so important to several areas of astronomy, are the
subjects of a recent (April 1993) Sky
and Telescope
article: "Getting The Measure Of Gravity", p. 28. The
current 'best-value' is G = 6.6726 x 10-11
(SI units).
Now
that we have examined the individual physical factors contributing to
the law of gravitation, let us look at the equations as a whole. If
you haven't already noticed, the right sides of both equations are
the same. That is, F1
= F2.
The force exerted by object 1 on object 2 is exactly the same, but
in the opposite direction, to that of object 2 on object 1. This
relationship is a manifestation of Newton's 3rd
law of motion.
To paraphrase: for every action there is an equal and opposite
reaction!
2.
Motion
Based
on the experiments of the Italian scientist Galileo Galilei some 50
years earlier, and upon his own intuition, Isaac Newton proposed his
other two laws of motion. The 1st
law of motion
defines what is meant by a 'force'. This law states that an object
at rest, or an object moving in a straight line at constant speed,
will forever retain this motion, unless acted upon by a 'force'. In
other words, to change either the speed or direction of motion of an
object requires action by a force. Such a change in the state of
motion is called an 'acceleration'.
The
quantitative relationship between force and acceleration is contained
in the 2nd
law of motion.
This law states that the product of the mass (m) times acceleration
(a) is equal to the sum of all of the forces (F) acting on an object.
Symbolically, F = ma. In the case of the gravitational force
between two objects, the respective relationships are:
We
have already noted that F1
= F2,
and therefore equation (2) shows that:
The
ratio of accelerations is inversely proportional to the ratio of
masses. In other words, the less massive object will be accelerated
to higher speeds than the more massive object. That is certainly
consistent with our own experience. If you put all of your strength
(force) into pushing a stopped automobile, you might be able to
accelerate it to a few miles per hour. However, the same force
applied to a much more massive dump-truck may not produce any motion
at all!
Now
let us combine equation (2) with the law of gravitation (1). If we
divide through by the mass common to both expressions for the force
on the same object, we get expressions for the accelerations. That
is,
We
note that the acceleration of an object depends upon the mass of the
other object, but not upon its own mass. Equation (3) is a
quantitative expression of Galileo's observation that objects of
different mass (or weight) dropped from some height (the Leaning
Tower of Pisa?) will accelerate toward the ground at the same rate.
That is, in a vacuum so that air resistance will not affect an
object's fall speed, a feather and a 10 ton boulder will reach the
ground at exactly the same time!
3.
Some Conclusions
The
law of gravitation (1) has revealed that a mass of (say) 100 kg is
attracted 'downward' by the Earth's mass with a force (weight) of 220
pounds. The Earth is also attracted 'upward' with the same force.
However, the Earth is so much more massive (about 6 x 1024
kg), the planet's acceleration in response to such a tiny force is
correspondingly minuscule. Equations (2) and (3), show that the
accelerating effect of the Earth's mass on the 100 kg object (or any
object) is, however, quite substantial. As we all know, a jump from
even a height of 10 feet can produce bone-jarring speeds at impact
with the ground.
The
acceleration of gravity (g) for objects at the Earth's surface is:
Here,
m is the Earth's mass and r the Earth's radius, and the numerical
values are given in the english and metric system of units. Using
the more familiar english value of g, a falling object (neglecting
air resistance) would be travelling at a speed of 32 feet/second
after 1 second, at 64 feet/second after 2 seconds, ... , 320
feet/second (28 miles/hour) after 10 seconds, and faster and faster!
To
conclude our discussion, we note that Newton's basic laws of motion
and gravitation have performed so many useful tasks on behalf of
astronomy and other sciences over the past 3 centuries. Newton
himself was able to prove Kepler's laws of motion which were based
entirely on observations of the planets, and to show that the same
law that governs a falling 'apple' at the Earth's surface also
controls the motion of the Moon about the Earth. In future articles
and talks at Club meetings, the present writer will use the concepts
discussed here as a starting point for exploring the motion of
objects subjected to mutual gravitational interaction.
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