Newton's Law of Universal Gravitation
Gravity is More Than a Name
The Apple, the Moon, and the Inverse Square Law
Newton's Law of Universal Gravitation
Cavendish and the Value of G
The Value of g
As discussed earlier in Lesson 3,
Isaac Newton compared the acceleration of the moon to the acceleration
of objects on earth. Believing that gravitational forces were
responsible for each, Newton was able to draw an important conclusion
about the dependence of gravity upon distance. This comparison led him
to conclude that the force of gravitational attraction between the Earth
and other objects is inversely proportional to the distance separating
the earth's center from the object's center. But distance is not the
only variable affecting the magnitude of a gravitational force. Consider
Newton's famous equation
Fnet = m • a
Newton knew that the force that caused the apple's
acceleration (gravity) must be dependent upon the mass of the apple. And
since the force acting to cause the apple's downward acceleration also
causes the earth's upward acceleration (Newton's third law), that force
must also depend upon the mass of the earth. So for Newton, the force of
gravity acting between the earth and any other object is directly
proportional to the mass of the earth, directly proportional to the mass
of the object, and inversely proportional to the square of the distance
that separates the centers of the earth and the object.
The UNIVERSAL Gravitation Equation
But Newton's law of universal gravitation extends gravity beyond earth. Newton's law of universal gravitation is about the universality of gravity. Newton's place in the Gravity Hall of Fame is not due to his discovery of gravity, but rather due to his discovery that gravitation is universal. ALL
objects attract each other with a force of gravitational attraction.
Gravity is universal. This force of gravitational attraction is directly
dependent upon the masses of both objects and inversely proportional to
the square of the distance that separates their centers. Newton's
conclusion about the magnitude of gravitational forces is summarized
symbolically as
Since the gravitational force is directly proportional to the mass of
both interacting objects, more massive objects will attract each other
with a greater gravitational force. So as the mass of either object
increases, the force of gravitational attraction between them also
increases. If the mass of one of the objects is doubled, then the force
of gravity between them is doubled. If the mass of one of the objects is
tripled, then the force of gravity between them is tripled. If the mass
of both of the objects is doubled, then the force of gravity between
them is quadrupled; and so on.
Since gravitational force is inversely proportional
to the square of the separation distance between the two interacting
objects, more separation distance will result in weaker gravitational
forces. So as two objects are separated from each other, the force of
gravitational attraction between them also decreases. If the separation
distance between two objects is doubled (increased by a factor of 2),
then the force of gravitational attraction is decreased by a factor of 4
(2 raised to the second power). If the separation distance between any
two objects is tripled (increased by a factor of 3), then the force of
gravitational attraction is decreased by a factor of 9 (3 raised to the
second power).
G = 6.673 x 10-11 N m2/kg2
The units on G may seem rather odd; nonetheless they are sensible.
When the units on G are substituted into the equation above and
multiplied by m1• m2 units and divided by d2 units, the result will be Newtons - the unit of force.
Using Newton's Gravitation Equation to Solve Problems
Knowing the value of G allows us to calculate the force of
gravitational attraction between any two objects of known mass and known
separation distance. As a first example, consider the following
problem.
Sample Problem #1
Determine the force of gravitational attraction between the earth (m = 5.98 x 1024 kg) and a 70-kg physics student if the student is standing at sea level, a distance of 6.38 x 106 m from earth's center.
The solution of the problem involves substituting known values of G (6.673 x 10-11 N m2/kg2), m1 (5.98 x 1024 kg), m2 (70 kg) and d (6.38 x 106 m) into the universal gravitation equation and solving for Fgrav. The solution is as follows:
Sample Problem #2
Determine the force of gravitational attraction between the earth (m = 5.98 x 1024
kg) and a 70-kg physics student if the student is in an airplane at
40000 feet above earth's surface. This would place the student a
distance of 6.39 x 106 m from earth's center.
The solution of the problem involves substituting known values of G (6.673 x 10-11 N m2/kg2), m1 (5.98 x 1024 kg), m2 (70 kg) and d (6.39 x 106 m) into the universal gravitation equation and solving for Fgrav. The solution is as follows:
