13.3 Newton’s Law of Gravity

Newton proposed that every object in the universe attracts every other object with a force that is

Inversely proportional to the square of the distance between the objects.
Directly proportional to the product of the masses of the two objects.

To make these ideas more specific, FIGURE 13.4 shows masses $m_{1}$ and $m_{2}$ separated by distance r. Each mass exerts an attractive force on the other, a force that we call the gravitational force. These two forces form an action/reaction pair, so $\vec{F}_{1 on 2}$ is equal and opposite to $\vec{F}_{2 on 1}$ . The magnitude of the forces is given by Newton’s law of gravity.

The figure illustrates the characteristics of the force vectors acting on two masses labeled m subscript 1 and m subscript 2.

FIGURE 13.4 Full Alternative Text

The constant G, called the gravitational constant, is a proportionality constant necessary to relate the masses, measured in kilograms, to the force, measured in newtons. In the SI system of units, G has the value

G = 6.67 \times 10^{- 11} {N m}^{2} {/kg}^{2}

FIGURE 13.5 is a graph of the gravitational force as a function of the distance between the two masses. As you can see, an inverse-square force decreases rapidly.

The figure illustrates a gravitational force versus Separation of masses graph.

FIGURE 13.5 Full Alternative Text

Strictly speaking, Equation 13.2 is valid only for particles. However, Newton was able to show that this equation also applies to spherical objects, such as planets, if r is the distance between their centers. Our intuition and common sense suggest this to us, as they did to Newton. The rather difficult proof is not essential, so we will omit it.

13.3 Newton’s Law of Gravity

FIGURE 13.4 The gravitational forces on masses $m_{1}$ and $m_{2}$ .

FIGURE 13.5 The gravitational force is an inverse-square force.