More Practice with Gravity

You will need to use your scientific calculator to check the math in the following examples. You may find, like so many students of introductory astronomy, that the hardest thing about exponents is figuring out which buttons to push on your calculator. For the calculator provided with Microsoft Windows, there is an "Exp" key. First, put in the base number (for example, in the number 6.67 ×10^–11, the "6.67" is the base number). Then click on the "Exp" key and enter the exponent, clicking on "+/–" to make the exponent negative if needed.

The Force of Gravity Between You and the Earth

The equation that we use for determining the force between the Earth and you is:

force =

where M₁ is the mass of Earth in kilograms; M₂, your mass in kilograms; and R, the radius of the Earth in meters. The G is the gravitational constant: 6.67 ×10^–11 N m²/kg². Let's not worry too much about the units of measure, however.

Examples

1. A realistic example. Your mass is 50 kilograms (70 kg * 2.2 lbs/kg = 154 pounds). The force felt between you and the Earth (equal and opposite) is:
   
   force =  =  = 686 N
   
   
   
2. Let's double the radius of the Earth,
   
   so that R = 1.276 × 10¹³ meters.
   
   force =  = 171.5 N
   
   you would weigh 1/4 as much!
   
   
   
3. Let's triple the radius of the Earth,
   
   so that R = 1.276 × 10¹³.
   
   force =  =  = 76.2 N.
   
   you would weigh 1/9 as much (686/9 = 76.2, all numbers are rounded off)!
   
   
   
4. How about a little trick? Let's say the Earth shrank, so that its radius becomes 1/2 of what it was before. Now, the girl, still standing on the surface, is closer to the center of the Earth. Before we go through the calculations, you should predict whether she will weigh
   1. the same,
   2. more, or
   3. less.

If you guess "more" or "less," then predict how much more or less.

The new radius of the Earth, after shrinking, is 3.19 × 10⁶ meters, so the new force becomes:

force =  = 2744 N

Let's see: 2744 divided by 686 = 4. If the Earth were to shrink to 1/2 of its current radius, you would weigh four times as much!

5.  Following this line of reasoning, if the Earth's radius were 1/3 of its current radius, you would weigh 9 times as much. (Remember when we are doing these calculations, the mass of you and the mass of the Earth are not changing.)
   
   The force between you and the Earth depends directly on the two masses. It doesn't take a rocket scientist to know that if you ate so much that you ended up with two times your current mass, you would weigh two times as much. The force between you and the Earth would double, and your ankles would ache.
   
   You will not be required to solve any equations involving such large quantities with exponents; however, you should have an understanding of how the force between you and the Earth would change should either or both of the masses change, or if the radius of the Earth changed.

The Force of Gravity Between Two Worlds

You may have wondered during the above examples why we always measured from the center of the Earth to the surface, in other words, used the radius of the Earth. That is because solid, spherical objects act as if all of the mass were contained right in the very center (this is not strictly true, but a good enough approximation for our use).

Let's take look at other examples, now involving two spherical worlds. This time let's use some values for the masses that makes our arithmetic a bit more manageable, and get rid of those pesky exponents. Because we want to look at relative changes, we are also going to ignore "Big G." The equation we use here is similar to the one above, but this time we substitute D, meaning the distance between the centers of the two worlds. The equation is

force = 

where this time the two masses, M₁ and M₂ represent the masses of the two worlds. The worlds are shown here all at approximately the same size, but the radii do not matter, just the distances between the centers.

Calculate the force between each of the worlds, substituting the values shown in the figure into the equation above. Do not bother with units of measure here, it's the concept we want to understand. Make sure you try each example before looking at the answers.