# The Law of Gravitation

When a person does a great deal of work in a scientific field, it often happens that that person eventually receives credit for almost everything done by anybody (see Matthew 13:12 in the Christian Bible). For instance, Newton is given credit, at least in popular accounts, for almost every interesting thing that happened in science during the Renaissance. In fact, then as now, science is the activity of a community, with many people contributing to the revolution in thinking. For example, the essentials of Newton's first law of motion were discovered by Galileo, and Robert Hooke surmised the essential parts of the law of gravitation.

Newton's genius lay in his ability to see a simple underlying law for very different phenomena and to synthesize diverse branches of science. An example of this is his realization that both the motion of the Moon and the motion of a falling apple could be explained by the same force, the force of gravity. In this chapter we will study the physics of the gravitational force.

Newton's law of gravity states that any two objects exert an attractive force on each other given by

Newton's genius lay in his ability to see a simple underlying law for very different phenomena and to synthesize diverse branches of science. An example of this is his realization that both the motion of the Moon and the motion of a falling apple could be explained by the same force, the force of gravity. In this chapter we will study the physics of the gravitational force.

Newton's law of gravity states that any two objects exert an attractive force on each other given by

(1)

Here

*F*_{grav }is the magnitude of the gravitational force between two objects,*m*_{1}and*m*_{2}are the masses of the objects,*d*is the distance between the centers of the objects, and*G*is a universal constant 6.67 x 10^{â€“11}m^{3}/kg s^{2}. Do not memorize*G*, but do remember the equation. We discussed it in Chapter 1.Example

What is the force on a cow (200 kg) standing on the surface of the Earth? (Assume *M*_{Earth }= 6.0 x 10^{24} kg, *R*_{Earth }= 6.4 x 10^{6} m.)

**Figure 4-1**

Solution

Let's assume we have a spherical cow (Figure 4-1). We calculate

Example

What is the motion of an apple (0.1 kg) which has let go of its tree?

Solution

First, we DRAW A DIAGRAM (Figure 4-2) showing all the forces on the apple while it is falling. There is only the force of gravity (nothing else is touching it), so we write

**Figure 4-2**

Since the total force is simply the gravitational force, we write

So the apple accelerates downward at 9.8 m/s^{2}.

Equation (1) is easy to use in two types of problems:

- obtaining the force between two planets (
*d*is much larger than the radii of the planets), and - obtaining the force between a planet and a small object on its surface (
*d*is essentially the radius of the planet).

The previous examples illustrated this second use.