


Newton’s Law of Universal Gravitation
Newton’s Law of Universal Gravitation is
a fundamental physical law. We experience its effects everywhere
on this planet, and it is the prime mover in the vast world of astronomy. It
can also be expressed in a relatively simple mathematical formula
on which SAT II Physics is almost certain to test you.
Gravitational Force
In 1687, Isaac Newton published his Law of Gravitation
in Philosophiae Naturalis Principia Mathematica.
Newton proposed that every body in the universe is attracted to
every other body with a force that is directly proportional to the
product of the bodies’ masses and inversely proportional to the
square of the bodies’ separation. In terms of mathematical relationships,
Newton’s Law of Gravitation states that the force of gravity, , between two particles of mass and has a magnitude of:
where r is the distance
between the center of the two masses and G is
the gravitational constant. The value of G was
determined experimentally by Henry Cavendish in 1798:
The force of gravity is a vector quantity. Particle attracts particle with a force that is directed toward , as illustrated in the figure below. Similarly,
particle attracts particle with a force that is directed toward .
Note that the gravitational force, , acting on particle is equal and opposite to the gravitational
force acting on particle , –. This is a consequence of Newton’s Third Law.
Let’s consider two examples to give you a more intuitive
feel for the strength of the gravitational force. The force of gravity
between two oranges on opposite sides of a table is quite tiny,
roughly 10^{–13} N. On the
other hand, the gravitational force between two galaxies separated
by 10^{6} light years is something
in the neighborhood of 10^{27} N!
Newton’s Law of Gravitation was an enormous achievement,
precisely because it synthesized the laws that govern motion on
Earth and in the heavens. Additionally, Newton’s work had a profound
effect on philosophical thought. His research implied that the universe
was a rational place that could be described by universal, scientific
laws. But this is knowledge for another course. If you are interested
in learning more about it, make sure to take a class on the history
of science in college.
Gravity on the Surface of Planets
Previously, we noted that the acceleration due to gravity
on Earth is 9.8 m/s^{2} toward
the center of the Earth. We can derive this result using Newton’s
Law of Gravitation.
Consider the general case of a mass accelerating toward
the center of a planet. Applying Newton’s Second Law, we find:
Note that this equation tells us that acceleration is
directly proportional to the mass of the planet and inversely proportional
to the square of the radius. The mass of the object under the influence
of the planet’s gravitational pull doesn’t factor into the equation.
This is now pretty common knowledge, but it still trips up students
on SAT II Physics: all objects under the influence of gravity, regardless
of mass, fall with the same acceleration.
Acceleration on the Surface of the Earth
To find the acceleration due to gravity on the surface
of the Earth, we must substitute values for the gravitational constant,
the mass of the Earth, and the radius of the Earth into the equation
above:
Not coincidentally, this is the same number we’ve been
using in all those kinematic equations.
Acceleration Beneath the Surface of the Earth
If you were to burrow deep into the bowels of the Earth,
the acceleration due to gravity would be different. This difference
would be due not only to the fact that the value of r would
have decreased. It would also be due to the fact that not all of
the Earth’s mass would be under you. The mass above your head wouldn’t
draw you toward the center of the Earth—quite the opposite—and so
the value of would also decrease as you burrowed. It
turns out that there is a linear relationship between the acceleration
due to gravity and one’s distance from the Earth’s center when you
are beneath the surface of the Earth. Burrow halfway to the center
of the Earth and the acceleration due to gravity will be
^{1}/_{2} g.
Burrow threequarters of the way to the center of the Earth and
the acceleration due to gravity will be ^{1}
/_{4} g.
Orbits
The orbit of satellites—whether of artificial
satellites or natural ones like moons and planets—is a common way
in which SAT II Physics will test your knowledge of both uniform circular
motion and gravitation in a single question.
How Do Orbits Work?
Imagine a baseball pitcher with a very strong arm. If
he just tosses the ball lightly, it will fall to the ground right
in front of him. If he pitches the ball at 100 miles
per hour in a line horizontal with the Earth, it will fly somewhere
in the neighborhood of 80 feet before it hits the ground.
By the same token, if he were to pitch the ball at 100,000 miles
per hour in a line horizontal with the Earth, it will fly somewhere
in the neighborhood of 16 miles before it hits the
ground. Now remember: the Earth is round, so if the ball flies far
enough, the ball’s downward trajectory will simply follow the curvature
of the Earth until it makes a full circle of the Earth and hits
the pitcher in the back of the head. A satellite in orbit is an object
in free fall moving at a high enough velocity that it falls around
the Earth rather than back down to the Earth.
Gravitational Force and Velocity of an Orbiting
Satellite
Let’s take the example of a satellite of mass orbiting the Earth with a velocity v.
The satellite is a distance R from
the center of the Earth, and the Earth has a mass of .
The centripetal force acting on the satellite is the gravitational
force of the Earth. Equating the formulas for gravitational force
and centripetal force we can solve for v:
As you can see, for a planet of a given mass, each radius
of orbit corresponds with a certain velocity. That is, any object
orbiting at radius R must be orbiting
with a velocity of . If the satellite’s
speed is too slow, then the satellite will fall back down to Earth.
If the satellite’s speed is too fast, then the satellite will fly
out into space.
