Professor Dave here, I want to tell you
about Newton's law of universal gravitation.
We learned about Newton's
laws of motion, but there's one more law
of his to discuss, and it's a big one.
It's Newton's law of universal
gravitation. In possibly one of the
greatest strokes of genius in the
history of mankind, Newton looked at the
motion of the planets in their nearly
circular orbits around the Sun and
understood that the centripetal force
causing this motion was precisely the
same force that causes objects to fall
down towards Earth, which we call the
gravitational force. In this way he
proposed that the planets are in a kind
of free fall towards the Sun just the
same way that the Apple that mythically
hit him on the head to produce this
insight was in free fall towards the
earth. The objects differ greatly in size
but the concept is the same. Newton
corroborated this notion with a thought
experiment. When a cannon fires a cannon ball,
the ball eventually hits the ground.
If another cannonball is fired with
greater force, it will go a little
further before hitting the ground. If a
cannonball could be fired with an
incredibly immense force, it could
produce a speed so great that the ball
would never hit the ground, since it
would fall at the same rate that Earth's
curvature is produced. It would thus
always be falling towards the earth but
never hitting it. Such an object would be
said to be in orbit around the Earth.
Of course no cannon can do this, but we have
finally achieved this feat with all of
our satellites and space stations, which
are brought up to orbit on rockets .These
are very far from Earth's surface and
they are moving with such great speed
that they, along with anyone on board, are
always falling towards the earth but
never hitting it, in a free fall just
like Newton's apple. This means they
orbit around the Earth indefinitely at a
fixed speed and radius. The same can be
said for all the planets around the Sun.
This gravitational force can describe
the motion of every object
in space and it is the case that every
object that contains mass will exert
gravity on every other massive object.
Of course to feel the effects of gravity we
must be near an enormous object, like a
planet, but it is completely accurate to
say that gravitational force is exerted
by your car, your refrigerator, even you
yourself. It is just that this force is
completely negligible compared to the
gravitational force exerted by the earth.
Newton developed an equation to quantify
the magnitude of the gravitational force
between two objects, and it looks like
this, where F is equal to the constant of
universal gravitation, G, times the mass
of the first object, times the mass of
the second object, divided by the
distance between them squared. This
constant, like any other constant, simply
exists so that a natural phenomenon like
gravity can be expressed in our own
arbitrary man-made units, and it is equal
to 6.67 times 10 to the negative 11
Newton meters squared over kilogram
squared. These are the units that will
cancel out the units on the masses and
radius so as to give a value for force
in newtons. This value is not known to
Newton at the time but was determined
experimentally about a hundred years
later by Henry Cavendish. When discussing
the radius between two objects we will
take the distance between their centers
rather than their surfaces, as Newton
showed that the gravitational force
exerted by an object depends only on its
mass and not on its volume, meaning that
when discussing gravity we can treat
everything as a point like mass. He had
to invent the calculus to do so, much to
the dismay of math students everywhere.
When examining a system like the earth
and the moon, we must understand that
both of these objects exert
gravitational force on the other, and
that these forces are equal in magnitude
meaning that both of these bodies rotate
around their combined center of mass, but
don't forget that F equals ma, so equal
forces will not produce equal
accelerations if the masses are
different. As it happens, the earth is
much much more massive than the moon so
the mutual gravitational force is able
to accelerate
the moon more than the earth, and the
center of mass for the system lies
within the earth itself, which is why we
simply observe the moon going around the
earth. The same can be said for Newton's
falling apple. The apple accelerates
towards the earth and the earth
accelerates towards the apple, but the
earth is more massive than the apple by
an inconceivable factor, so the
acceleration of the earth is not even
measurable whereas we can visually
confirm the acceleration of the apple.
Furthermore, we want to understand that
an apple will fall to the earth with the
same acceleration as a bowling ball or
any other massive object, if we disregard
wind resistance. Although
counterintuitive to some, we can
rationalize this if we understand that
while the force of gravity is able to
impart greater acceleration on a more
massive object, the more massive object
also has greater inertia or resistance
to being accelerated, so the end result
is that all objects accelerate towards
Earth in the same way, at 9.8 meters per
second squared. This fact is easy to
derive if we do some algebraic
manipulation. We know that a falling
object exhibits behavior according to
Newton's second law, F equals ma, where
the force that generates the falling is
equal to the mass of the object times
its acceleration, but this force is the
gravitational force, so we can also model
the falling behavior with G m1 m2 over r
squared, where m 1 is the mass of the
object and m2 is the mass of the earth.
If we set these equal to each other, the
mass of the object is found on both
sides and will cancel out, so we can see
that the acceleration due to gravity is
equal to the gravitational constant
times the mass of the Earth divided by
the radius squared. This means that the
mass of an object does not affect the
rate of free fall.
Newton's work on gravity
was revolutionary. It correlated an
incredible amount of data, from
terrestrial motion to celestial motion,
which is all any good theory can hope to
do. But he could not explain how objects
can exert the gravitational force on one
another from a distance. Later, scientists
solved this problem by labeling gravity
as a field force, stating that matter
generates gravitational fields in space.
This was a bit more satisfactory, but it
wasn't until Einstein's general theory
of relativity that we arrived at a more
sophisticated understanding of gravity,
which has helped us learn about the
structure of space itself, as well as how
planets and stars and galaxies form. We
are still trying to fully understand
gravity today, but the continuation of
this discussion will have to wait until
the modern physics course, so for now
let's check comprehension.
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