Newton’s Law of Gravitation

Standards:

PH1 m. * Students know how to solve problems involving the forces between two electric charges at a distance (Coulomb’s law) or the forces between two masses at a distance (universal gravitation).

PH1 e. Students know the relationship between the universal law of gravitation and the effect of gravity on an object at the surface of Earth.

Lesson Objective: To be understand the ideas of “Action at a Distance”, the 1/r squared law, and to be able to calculate the gravitational force between two massive objects.

Newton knew that if one object pushes on another or if there was a rope between 2 objects and one pulled the other, the object would experience an acceleration.

Newton was fascinated that the earth could pull on objects without any direct connection. He called this effect “Action at a Distance”.

Newton believed that objects had some internal essence, related to their mass that somehow spread out through space and got weaker as objects were moved farther from each other. He checked his idea by considering the orbital radii and speeds of the moons of Jupiter as compared with the Earth’s moon. This all made sense.

First understand how the surface of a sphere grows as the radius increases:

Sphere-math

Draw this diagram in your notebooks and identify (label)
a] the circumference of the circle
b] the area of the circle
c] the surface area of the sphere
d] the volume of the sphere

Now consider that the gravitational essence of a mass at the center of the sphere spreads out evenly to reach the surface of a sphere 1 meter in radius.

What happens if the radius of the sphere increases to 2 meters? (discuss)

From this Newton realized that the effect of gravity must get smaller by a factor of 1/r^2.

Now let’s use Newton’s 2nd Law to develop a formula for the force of gravity.

F=ma

F=mg

g (the acceleration due to gravity at the earth’s surface) must depend on the mass of the earth (M) and the radius of the earth squared. There also needs to be some constant of proportionality – Newton called this constant “G” The Universal Gravitational Constant.

F= GmM/r^2

The actual value of G remained a mystery until it was determined experimentally by Cavendish 120 years later in 1798.

G = 6.67 x 10^-11 N.m^2/kg^2