MadSci Network: Astronomy Query:

### Re: what is the acceleration of the moon?

Date: Sun Nov 21 11:31:25 1999
Area of science: Astronomy
ID: 938149636.As
Message:

A fundamental principle in physics, first formulated by Isaac Newton, is that, objects in motion remain in motion, moving in a straight line, unless acted upon by an external force. Thus, in order for the Moon to continue revolving or remain in orbit around the Earth, rather than flying off in a straight line, it must be acted upon by an external force. That force is the Earth's gravity.

The force of the Earth's gravity causes the Moon to accelerate toward the Earth. Fortunately, the Moon is also moving transversely to the line joining the Moon and the Earth. Thus, the Moon never impacts the Earth, but continually "falls around" it.

The acceleration of the Moon can be computed using another two principles formulated by Newton,

```         G*M*m
ma = F = -----.
d*d
```
Here I've equated the force due to the gravity between the Earth and Moon [`G*M*m/(d*d)`] to the force on the Moon because of the Earth's acceleration (`ma`). The Moon's mass is `m`, the Earth's mass is `M`, the distance between the two bodies is `d`, and `G` is a constant known as Newton's constant of gravitation.

Solving for the accleration on the Moon, we find

```    G*M
a = ---.
d*d
```
One important aspect of the force of gravity is already apparent. The accleration of an object due to the force of gravity is independent of its mass. If you were in orbit about the Earth, and at the same distance as the Moon, you would experience exactly the same amount of acceleration as the Moon does. (This fact seems strange to us on the Earth. After all, a bowling ball and a feather do not fall at the same rate toward the Earth; the bowling ball will hit ground before the feather does. However, there's another force acting near the Earth's surface---air resistance. In fact, the Apollo astronauts did exactly this experiment on the surface of the Moon, and the two objects hit the Moon at the same time.)

Numerically, we can use measured values to find `a`. Using ```m = 59.8 x 1023``` kilograms, `d = 3.75 x 108` meters, and `G = 6.67 x 10-11` N m2/kg2, we find

```a = 0.0028 m/s2
```
for the acceleration of the Moon by the Earth's gravity. (I've used a "round" number for the distance between the Earth and Moon. The Moon's orbit is not a perfect circle, so its accleration varies slightly over the course of a month.) For comparison, the acceleration at the Earth's surface is 9.8 m/s2.

Exercise for the reader: If the Moon's mass is ```0.7 x 1023``` kg, what is the Earth's acceleration toward the Moon?

Current Queue | Current Queue for Astronomy | Astronomy archives