Why is force defined as the rate of change of momentum?

[Ryder Rude]

How did this definition come to Newton's mind? Force seems to be the effort we apply. So, qualitatively, more the effort, more the force. How to understand intuitively that this formula given by Newton gives a quantitative measure of Force?
And, how do we know that Force does not depend on more variables? I think an infinite number of formulas can be given to measure the 'effort' we make. For example, I could give a formula to measure roughness of a surface and could define the unit force as the amount of force required to move a body of unit mass through a distance of 1m on a surface of unit roughness. I think that this definition could also measure the effort. Also, why not |m|+|a| or |m|^|a|. Why this particular definition of force?

 

[A.T.]

I think an infinite number of formulas can be given to measure the 'effort' we make.

That's exactly why your vague concept of "effort" is not useful. Momentum on the other hand, is a conserved quantity, and therefore useful to make quantitative predictions.

 

[PeroK]

How did this definition come to Newton's mind? Force seems to be the effort we apply. So, qualitatively, more the effort, more the force. How to understand intuitively that this formula given by Newton gives a quantitative measure of Force?
And, how do we know that Force does not depend on more variables? I think an infinite number of formulas can be given to measure the 'effort' we make. For example, I could give a formula to measure roughness of a surface and could define the unit force as the amount of force required to move a body of unit mass through a distance of 1m on a surface of unit roughness. I think that this definition could also measure the effort. Also, why not |m|+|a| or |m|^|a|. Why this particular definition of force?

It's a characteristic of scientific thought to see through the variations and complexities to the general underlying relationships.

For example, you could apply your method to gravity and conclude that gravity works in all sorts of different ways in different circumstances to explain, for example: a heavy object sometimes falling to Earth and sometimes flying (an aeroplane); why gravity doesn't act on a boat at sea and cause it to sink; and why very light objects float around seemingly at random.

The insight brought by scientific thought is to see that gravity is doing the same thing in all cases, but the different circumstances are caused by the complexities of the medium (air or water). And that if you could remove the medium then all objects would fall at the same rate.

That was not immediately obvious and it took a great scientific mind to understand it for the first time.

The complexities over force are similar. It takes great scientific insight to get to $F = ma = \frac{dp}{dt}$ and not get bogged down in all the different ways a force can act and look for different laws for each.

 

[Kumar8434]

How did this definition come to Newton's mind? Force seems to be the effort we apply. So, qualitatively, more the effort, more the force. How to understand intuitively that this formula given by Newton gives a quantitative measure of Force?
And, how do we know that Force does not depend on more variables? I think an infinite number of formulas can be given to measure the 'effort' we make. For example, I could give a formula to measure roughness of a surface and could define the unit force as the amount of force required to move a body of unit mass through a distance of 1m on a surface of unit roughness. I think that this definition could also measure the effort. Also, why not |m|+|a| or |m|^|a|. Why this particular definition of force?

Let's talk about your definition of force, which appears to be $mass*distance moved*roughness of surface$ . Apply the same 'effort' on two blocks of same mass on two surfaces of same roughness, one on Earth and one on the Moon, the same block will move through different distances. So, this definition will give different numerical values of the same force. Also, the block may not move through any distance even if you apply some effort on it because of non-linear nature of friction. This definition won't work.
Now, for your third question: why not $|m|+|a|$ or $|m|^{|a|}$? I think the technical answer to this question is related to dimensions but I think this is a better explanation of 'why $m\cdot a$?'
Short answer: Our minds are linear. They seem to connect better with multiplicative relations.
Explanation: Apparently, humans have a better intuition of multiplication than of addition and definitely better than that of exponentiation. Ask why $speed=\frac{distance}{time}?$
Let's say you've to give a formula which gives you the numerical value of speed. First, guess that on what quantities speed might depend on. The answer seems to be the distance moved and the time taken to move that distance. Now, intuitively we can work out the requirements of our formula:
1. If A has moved more distance than B in the same time, then our formula should give a higher numerical value for the speed of A than the speed of B.
2. If A takes less time than B to move the same distance, then our formula should give a higher numerical value of the speed of A than the speed of B.
Now, we could have these formulas: $distance-time$, $\frac{distance}{time}$ and $\log_{time}{distance}$, ${distance}^2-{time}^2$, etc.
Now, suppose A moves $3m$ in $3s$ and B moves $6m$ in $3s$. Here are the numerical values of speeds of A and B given by these four formulas:
1. A=3, B=0
2.A=2, B=1
3.A=1, B=1.63
4.A=27, B=0
Now, to which of these values of speeds your mind seems to connect to the most? Note that A moved twice distance than B in the same time, then wouldn't it make sense if the speed of A was also twice the speed of B? If A had moved 10 times the distance moved by B in the same time, then wouldn't it make sense if the speed of A was also 10 times? Only the second formula satisfies that. We seem to intuitively connect to multiplicative relations.
So, we use $\frac{distance}{time}$ for speed.
For a similar reason, the $\frac{dp}{dt}$ definition of force is more intuitive than others.

 

[mastermechanic]

I'm going to write more basic explanation. Newton says the best way to research or observe the nature is experiment, so he generally didn't define the formulas only by thinking what should be added to this formula. He always experimented actions, for example, he stretched two rubbers in the same distance and put different masses in front of them,so he observed that their accelerations were inversely propotional to their masses and because he didn't find another variable, he understood that the only varibles were "m" & "a" and he formed F=m.a Basically it is, don't confuse yourself just think simple, keep simple. KISS :)