# Understanding what it means that gravitation is proportional to the product of the masses

Newton's gravitational law states that $F = G*\frac{M*m}{d^2}$

Intuitively it means that the greater the masses, the stronger the force, but it is more precise than that, it is proportional to the product of the masses not, for example, their sum.

I am not sure how Newton derived that, but I can guess he deduced that if $M_1 = 2*M$, then the force on $m$ should be twice as much.

However, the question is: what if I was able to move mass away from $M$ and add it to $m$? The total mass of the two bodies did not change, but now the force with which they are attracted to each other has changed.

Is there an intuitive explanation of what is going on?

The most intuitive way of thinking of it seems to be to picture two objects each made up of 100 "particles" (imagine indivisible points of equivalent and constant mass) at a given separation $d$. We can build up to this picture and see why the product of masses makes sense.

First, imagine the simplest picture: 2 particles, $A$ and $B$ at separation $d$. Let's say that $A$ experiences 1 unit of force due to $B$ and $B$ experiences one unit of force due to $A$. Now let's add another particle $C$ that is right next to $B$. The gravitational attraction between $B$ and $C$ is very large but we can imagine it being canceled by some repulsive force of the particles touching. Now $A$ experiences 2 units of force, 1 from $B$ and 1 from $C$. The system of $B$ and $C$ together experiences 2 units of force, 1 from $B$ with $A$ and 1 from $C$ with $A$. So we can see how each particle in a system independently experiences gravitational attraction from each individual particle in the other system.

Now imagine the two objects ($X$ and $Y$), each with 100 particles stuck together, and the two objects at separation $d$. Now each particle in $X$ experiences attraction from every particle in $Y$ (and vice versa). So 100 partices in $X$ each individual experience 100 separate attractions, each of 1 unit of force. Therefore the total attraction is $100\times100=10000$ units of force.

Finally imagine $X$ has 50 particles and $Y$ has 150. The total mass of the two objects is the same as the previous case, but by the same argument $X$ has 50 particles which each independently experience attraction with 150 particles (with each instance of attraction imparting 1 unit of force), therefore $X$ experiences $150\times50=7500$ units of force.

Hopefully it is now more clear why force goes with the product of masses.

Perhaps it will help to look at the gravitational field strength, which here is measured in units of acceleration ($m/s^2$). Of course we know that force, mass and acceleration are related by $F=m\cdot a$. And we have Einsteins principle that an acceleration and a gravitational field are indistinguishable.

So, the gravitational field of the bigger body will be $g = G\cdot \frac{M}{r^2}$. The smaller mass (m) then experiences the force $F=m\cdot g$. Since M and m are the only two objects here, they both experience the same strength force, just in opposite directions.

That is the reason why the force between two gravitating Newtonian objects is proportional to their mass. You can swap the two masses m and M and arrive at the same result.

Also, consider a limiting case in which $m\rightarrow 0$. The force between the two bodies vanishes. That's not surprising. If there is no second object, there won't be a force on the first.