I was in physics class and we were talking about the gravitational constant G (6.67 x 10^-11 Nm^2/Kg^2). The question came up:
"Why does $F= (GMm)/r^2$ have a constant of proportionality and not $F=ma$?"
Our teacher said that it had to deal with the way we define our base unit of measurement for force, the Newton, but didn't explain it much beyond that.
The following is my solution to the question. Please tell me if you agree with my logic.
In physics, we first realized force was proportional to the multiplcation of mass and acceleration so we decided to measure force this way. We defined our basic force unit, the Newton, so that 1 newton is the force that causes a 1 kg object to accelerate at 1m/s^2. When we plug these numbers into our proportionality for force, F=kma, and solve for k, k=1, and the equation turns to f=ma.
We also realize that there are other proportionalities for force, for example: F is proportional to (Mm)/r^2 or F is proportional to (Qq)/r^2. These can be rewritten in the form F=(kMm)/r^2 and F=(kQq)/r^2.
In order to solve for each of these constants of proportionality, we need to have one number for the right hand term and one for the left.
For example: When we measure the force of gravity between two 1 kg objects 1 m apart, the resulting force is 6.67 x 10^11 times smaller than the force accelerating a 1 kg object at 1 m/s^2. This is just the nature of the universe. Different types of forces have different strengths. If we called the latter force 1 newton, like we've defined it, then the gravitational force is 6.67 x 10^-11 N. Thus, in F=(kMm)/r^2, k must have a value of 6.67 x 10^-11.
However, we could've defined our units for force in terms of gravity. If we called the force of grav. between two 1kg objects 1m apart a Newton, then in this situation, G would equal 1.
This would also change our constants of proportionality in other equations. For example, F=ma would now be F=6.67 x 10^11 ma. This is because the force accelerating a 1 kg object at 1m/s^2 is 6.67 x 10^11 times greater than our gravitational force in the previous example which we named a newton. Therefore, using our new definition of the newton, this applied force is 6.67 x 10^11 N and k in f=kma must = 6.67x10^11.
From these examples, I draw some conclusions:
The way we define our unit of force determines our constants of proportionality in all our relations for force. Additionaly, changing the units we measured in would also change our k values. This would give us different values for the right and left hand terms of our proportionalities and thus k would change in those proportionalities.
Does my logic seem correct? I also have one more question: can we generalize the previous statement? Is it fair to say that the way we define any unit of measurement for any physical quantity determines the k values in all the equations for that physicaly quantity?