Question Video: Identifying the Correct Formula for Newton’s Law of Gravitation Physics

Video Transcript

Which of the following is the correct formula for Newton’s law of gravitation? (A) 𝐹 is equal to 𝐺 times 𝑚 one times 𝑚 two divided by 𝑟 cubed. (B) 𝐹 is equal to 𝐺 times 𝑚 one times 𝑚 two divided by 𝑟 squared. (C) 𝐹 is equal to 𝐺 times 𝑚 one times 𝑚 two divided by 𝑟. (D) 𝐹 is equal to 𝐺 times the quantity 𝑚 one plus 𝑚 two all divided by 𝑟 squared. (E) 𝐹 is equal to 𝐺 times the quantity 𝑚 one plus 𝑚 two all divided by 𝑟.

The first thing that we can recall about Newton’s law of gravitation is that this is an example of an inverse square law. In this case, this means that the gravitational force 𝐹 between two objects is proportional to the inverse of the square of the distance between those two objects’ centers of mass. So, if we had masses 𝑚 one and two here, the gravitational force between them would depend on the distance between their centers, 𝑟. And specifically, it will be inversely related to that distance squared.

This means that any of our answer options that do not have 𝑟 squared in the denominator can’t be correct. For example, option (A) has 𝑟 cubed in the denominator. That doesn’t represent an inverse square law, likewise for answer options (C) and (E). Neither of these has 𝑟 squared in the denominator. The difference between our two remaining answer choices is that one of them multiplies the masses and the other adds them together.

To consider which one of these is correct, let’s say we take our object with mass 𝑚 one and we double that mass so that now its mass is two times 𝑚 one. This would have an effect of doubling the overall gravitational force between these masses. Similarly, if our second object, with mass 𝑚 two, were to suddenly lose half its mass, so it had a new mass value of 𝑚 two divided by two, then compared to the previous gravitational force between these masses, the new gravitational force would be one-half that old force.

For changes in mass like these to lead to such changes in the gravitational force between the masses, it’s necessary that in the formula for Newton’s law of gravitation the masses be multiplied together. This means we can now choose between options (B) and (D). The correct formula for Newton’s law of gravitation is that 𝐹 is equal to 𝐺, the universal gravitational constant, times 𝑚 one times 𝑚 two divided by 𝑟 squared. We choose answer option (B).