13.3.1 Gravitational Force and Weight

Knowing G, we can calculate the size of the gravitational force. Consider two 1.0 kg masses that are 1.0 m apart. According to Newton’s law of gravity, these two masses exert an attractive gravitational force on each other of magnitude

\begin{array}{l} F_{1 on 2} & = F_{2 on 1} = \frac{G m_{1} m_{2}}{r^{2}} \\ = \frac{(6.67 \times 10^{- 11} {N m}^{2} {/kg}^{2}) (1.0 kg) (1.0 kg)}{{(1.0 m)}^{2}} = 6.67 \times 10^{- 11} N \end{array}

This is an exceptionally tiny force, especially when compared to the gravitational force of the entire earth on each mass: $F_{G} = m g = 9.8 N$ .

The fact that the gravitational force between two ordinary-size objects is so small is the reason we are not aware of it. You are being attracted to every object around you, but the forces are so tiny in comparison to the normal forces and friction forces acting on you that they are completely undetectable. Only when one (or both) of the masses is exceptionally large—planet-size—does the force of gravity become important.

We find a more respectable result if we calculate the force of the earth on a 1.0 kg mass at the earth’s surface:

\begin{array}{l} F_{earth on 1 kg} & = \frac{G M_{e} m_{1 kg}}{R_{e}^{2}} \\ = \frac{(6.67 \times 10^{- 11} {N m}^{2} {/kg}^{2}) (5.97 \times 10^{24} kg) (1.0 kg)}{{(6.37 \times 10^{6} m)}^{2}} = 9.8 N \end{array}

where the distance between the mass and the center of the earth is the earth’s radius.

The earth’s mass $M_{e}$ and radius $R_{e}$ were taken from Table 13.2 in Section 13.6. This table, which is also printed inside the back cover of the book, contains astronomical data that will be used for examples and homework.

A photograph of the Andromeda galaxy. — A galaxy of $\approx 10^{11}$ stars spanning a distance greater than 100,000 light years.

Full Alternative Text

The force $F_{earth on 1 kg} = 9.8 N$ is exactly the weight of a stationary 1.0 kg mass: $F_{G} = m g = 9.8 N$ . Is this a coincidence? Of course not. Weight—the upward force of a spring scale—exactly balances the downward gravitational force, so numerically they must be equal.

Although weak, gravity is a long-range force. No matter how far apart two objects may be, there is a gravitational attraction between them given by Equation 13.2. Consequently, gravity is the most ubiquitous force in the universe. It not only keeps your feet on the ground, it also keeps the earth orbiting the sun, the solar system orbiting the center of the Milky Way galaxy, and the entire Milky Way galaxy performing an intricate orbital dance with other galaxies making up what is called the “local cluster” of galaxies.