To understand these kinds of problems, you need to remember two things about gravity:

1. Gravitational force is proportional to mass. In other words, if I increase mass, gravitational force increases. More specifically, if I multiply the mass by 3, for example, the gravitational force also gets multiplied by 3. If I divide the mass by 3, the gravitational force also gets divided by 3.

2. Gravitational force is inversely proportional to the distance squared. In other words, if the distance is increased, gravitational force DECREASES. More specifically, if I multiply the distance by 3, the gravitational force is DIVIDED by 3 squared, or 3×3, which is 9. If I divide distance by 3, gravitational force gets MULTIPLIED by 9.

Now, let’s look at the first part of Example 11.1:

“The gravitational force between two objects separated by a distance of 3 centimeters is measured. The objects are then brought closer together so that the distance between them is only 1 centimeter. What is the gravitational attraction now compared to when it was first measured?”

When we look at these problems, we first have to decide was the distance multiplied or divided? Well, the distance started out at 3 centimeters and then was changed to 1. How do you turn 3 into one? You divide by 3. Thus, the distance was divided by 3. When distance is divided by 3, the gravitational force is MULTIPLIED by 3×3. Thus, the gravitational force is multiplied by 9. That’s the same thing as saying that the gravitational force increases by a factor of 9.

What about the next one:

“The gravitational force between two objects separated by a distance of 5 centimeters is measured. Both objects are then replaced. The first object is replaced with one that has half of its mass, and the second object is replaced by one that has 8 times its mass. What is the gravitational attraction now compared to when it was first measured?”

Once again, we have to decide what happened. In this case, only the masses changed. I know that the distance is given, but it is not changed at all. Since the distance doesn’t change, we don’t have to mess with it. What else changed? One mass was replaced with something half its mass. In other words, the mass was DIVIDED by 2. That means you also divide the gravitational force by 2. The second mass was replaced with something 8 times as heavy. Thus, we multiplied the mass by 8. That means we multiply the gravitational force by 8 as well.

So…there are two changes. One divides the gravitational force by 2, and the other multiplies it by 8. Thus, the final gravitational force is equal to the original gravitational force divided by 2 and then multiplied by 8. If I take a number, divide by 2, and then multiply by 8, what is the next result? The net result is that I multiplied by 4. Thus, the new gravitational force is 4 times the original one.