Using Gravitation Relationships

Guide 12-2. Using Gravitation Relationships

The following table gives the primary relationships introduced in Chapter 12 and used in gravitation and satellite problems.

Eq. 1 is fundamental, while the other equations can be derived from these fundamental relationships: F_net = ma, a = v²/r, and v = d/t. Students must be able to derive Eqs. 2-4.

	Formula	Description	Method of Derivation
Eq.1		Gravitational force between two objects of mass m and M and whose centers of mass are separated by distance r	none
Eq.2		Acceleration of an object falling, rising, or in orbit around a planet or star of mass M at distance r from the center of the planet or star	See G12-1.
Eq.3		Speed of a satellite in circular orbit around a planet or star of mass M at distance r from the center of the planet or star	Set the net force on the satellite, F_net = mv²/r, equal to the gravitational force, F_g = GmM/r² and solve for v.
Eq.4		Period of a satellite in orbit around a planet or star of mass M at mean distance r from the center of the planet or star	Substitute v = 2πr/T for v in Eq. 3 and solve for T.

One way of doing calculations using the relationships above is to substitute known values of G, masses, and radii and simply calculate the results. Another method is to compare quantities by forming ratios and proportions. In the latter method, constants divide out and need not be substituted. This method is described in the guide Solving Gravitation Problems Using Proportional Reasoning. More generally, here's how the method works.

Decide what parameters are constant for the problem situation of interest.
Rewrite the equation as a proportion. This allows you to leave out the constants, since they will divide out.
Solve for the unknown.

Here's an example. Consider the system shown to the right of two satellites orbiting a planet. Let's compare forces, accelerations, speeds, and periods. Define these symbols:

m_o, m_i represent the masses of the blue and red satellites. (o = outer, i = inner)
M_p represents the mass of the planet.
r_o, r_i represent the radii of the satellites' orbits.

Suppose we're given that r_o/r_i = 2 and m_o/m_i = 3.

How do the accelerations of the satellites compare? The equation that applies is Eq. 2 above. In addition to G, M will be a constant, since M represents the mass of the planet, which is the same for both satellites. Therefore, we set up a proportion between acceleration and orbital radius.

Note that the radii are squared and are inverted compared to the accelerations. This, of course, is the nature of an inverse-squared relationship. Substituting the inverse ratio of the radii and squaring it, a_o/a_i = 1/4. It makes sense that the satellite which is further away experiences lesser acceleration.

Let's suppose now that you want to compare gravitational forces between the satellites and the planet. Now we use Eq. 1. We have the same constants G and M as before. Our proportion is:

Note that the force depends directly on the mass. Since m_o/m_i = 3, then F_o/F_i = 3/4.

Now compare orbital speeds using Eq. 3. Once again, the constants are G and M. Our proportion is:

We substitute 1/2 for the inverse of the given ratio of the orbital radii and take the square root to obtain v_o/v_i = sqrt(1/2). The outer satellite moves slower as expected.

Finally, we compare periods. We have the same constants as before. The proportion is:

We substitute r_o/r_i = 2, cube that, and take the square root. The result is T_o/T_i = sqrt(8) or 2sqrt(2). The outer planet has the greater period.