# Are parts of objects part of that object's mass?

Disclosure: I'm not a physicist or a scientist, so there is a chance that this will seem like a silly question for some of you. But I'm really curious about this and I want to know the answer.

According to Wikipedia, the formula which determines the gravitational force between 2 objects is:

$$F = G \times \frac{m1 \times m2}{r^2}$$

which means the force depends on the masses of the 2 objects.

But I've always wondered how exactly we determine the mass of an object, given that it can be easily divided into smaller parts.

The best example I can think of, in order to explain what I mean, is the Earth. We need a pretty exact mass In order to correctly and precisely calculate the orbit, how the Milankovitch cycles influence every movement of the planet and so on. I mean, I bet it's much more complicated than simply applying the above formula a few times, but I'm sure it's still extremely relevant.

So how do we calculate the mass ? What do we include in it ?

• Do we also count the mass of all the humans to be part of the mass of the Earth ?
• What about the mass of all the trees ?
• What about the mass of the mountains ?
• What about the mass of the atmosphere ?
• What about all the water in the oceans ? (about 0.02 % of the total Earth mass)

The objects I mentioned above are kind of irrelevant compared to the scale of the planet, but when you put them all together, the resulting number is not negligible. But all that is not the main point of my question.

What I want to know is: are they part of the object known as planet Earth ? If yes, where do you draw the line between objects that are part of the planet and objects that are not ?

What if there would be 2cm of vacuum between Earth's core and the layers above it ? Do you count all of it as just 1 object ?

• It really just depends on what you're looking to solve. If you want to calculate the moon's orbit, you include your mass because there's no way to distinguish you from the Earth at that scale. If you want to calculate how much energy an elevator needs to lift you from the Earth, you don't include your mass as part of Earth's mass, obviously. – Asher Jan 7 '16 at 20:55
• You may want to look at the Laws of Conservation too. I am guessing you may be hung up that the earth mass changes from "Lets say a forest growing" for an example. – Ed Yablecki Jan 7 '16 at 21:22
• Have you read my answer? – Arturo don Juan Jan 9 '16 at 22:00

## 2 Answers

If you have a compound object consisting of many interacting parts, there is a notion of the total mass (the sum) and there is even an equation of motion for this mass called the equation of motion of the center of masses: $M_{\rm{tot}}\mathbf{\ddot{R}}=\mathbf{F}$. Generally the force in this equation is not reduced to $\mathbf{F}(\mathbf{R})$, but to a force including the relative coordinates of parts of the whole body. In a specific case of uniform external (different) forces acting on body parts the equation force is a constant. To draw the line, one has to analyze simplifications the force $\mathbf{F}$ (gradients, etc.).

The formula you put there is the gravitational force between two point masses. This formula works to a high-degree of accuracy when you're measuring the gravitational force between two objects with a high-level of spherical symmetry and/or small-size relative to the distance between their center-of-masses. If you use that formula to calculate the mass of the Earth by dropping a pebble near the surface of the Earth, you'll technically be partly measuring the gravitational forces between the pebble and the trees, people, atmosphere, other planets, and the rest of the universe. Of course, those terms (the gravitational forces felt between the pebble and those objects) will turn out to be pretty negligible (as humans have defined that term) compared to the gravitational force between the pebble and the approximately spherical Earth, and so to a certain degree of accuracy, we will not be measuring them with that formula.

Once again, those terms will technically still be in that formula, so if you want to measure the mass of the "bulk of the Earth", you'll have to come up with nifty ways of measuring the gravitational forces felt between the pebble and the trees, people, atmosphere, moon, sun, other planets, ... you get the idea. You'll have to complicate your formula so as to rule out those extraneous contributions. Also, the Earth isn't perfectly spherical - it's an oblate spheroid, so you have to complicate your formula even further to include that.

It all comes down to how accurate (how many decimal places) you want your answer. If you only care about getting the first two digits, whether or not you consider the oceans, trees, atmosphere, and all that other stuff won't matter. If, however, you want it accurate to six or seven decimal places, you'll have to complicate your formula so as to correctly consider their contributions.