Question

Which of the following would cause the force exerted on the Moon by Earth's gravity to increase by the largest amount?

a. Double the mass of the Moon.
b. Double the mass of Earth.
c. Move the Moon two times farther away from Earth.
d. Move the Moon two times closer to Earth.

Answer 1

Move the Moon two times closer.

Step-by-step explanation:

Let
`M` denote the mass of the Earth, and let
`m` denote the mass of the Moon. Let
`r` denote the distance between the Earth and the Moon.

Let
`G` denote the Gravitational constant. The gravitational attraction between the Earth and the Moon would be:

`\begin{aligned}F &= (G\, M\, m)/(r^(2))\end{aligned}`.

Rewrite this equation to isolate
`m`, the mass of the Moon:

`\begin{aligned} F &= \left((G\, M)/(r^(2))\right) \, m\end{aligned}`.

In other words, if all other quantities stay the same, the magnitude of gravitational attraction between the Earth and the Moon is proportional to the mass of the Moon. Doubling the mass of the Moon would double the magnitude of this force.

Similarly, rewrite this equation to isolate
`r`, the distance between the Earth and the Moon:

`\begin{aligned} F &= (G\, M\, m)\, (1)/(r^(2))\end{aligned}`.

In other words, the magnitude of this force is inversely proportional to the squared distance between the Earth and the Moon. Reducing the distance to
`(1/2)\, r` would quadruple (
`(1 / (1/2))^(2)`) the magnitude of this force.

Therefore, among the options, moving the Moon two times closer would increase the magnitude of this force by the largest amount.