# Why doesn't Newton define F=m(a^2)

russ_watters
Mentor
If the claim was that "falling unequal masses accelerate at the same rate", you don't need to measure the force to prove it, you just need to simultaneously drop several objects of different weight/mass and see if they hit the ground at the same time. Which he is believed to have done, though the specifics of the experiments are in doubt: http://en.wikipedia.org/wiki/Galileo_Galilei#Falling_bodies
http://en.wikipedia.org/wiki/Inertia#Classical_inertia

How far he took the logic I'm not sure, but if we know acceleration due to gravity is constant, then a=g=f/m

If the claim was that "falling unequal masses accelerate at the same rate", you don't need to measure the force to prove it, you just need to simultaneously drop several objects of different weight/mass and see if they hit the ground at the same time. Which he is believed to have done, though the specifics of the experiments are in doubt: http://en.wikipedia.org/wiki/Galileo_Galilei#Falling_bodies
http://en.wikipedia.org/wiki/Inertia#Classical_inertia

How far he took the logic I'm not sure, but if we know acceleration due to gravity is constant, then a=g=f/m
But in order to make the proportion $\frac{F_1}{m_1}\textit{=}\frac{F_2}{m_2}$, as mikelepore stated, wouldn't force need to be quantified in some way? If you don't need to measure the forces, how did Galileo know to attribute the accelerations to the forces?

I'm not sure about Galileo's time, but after Newton people were able to say:

a1=a2 empirically from Galileo dropping things.
tentatively suppose a=F/m
F1/m1 = F2/m2
substitute the law of universal gravitation
(G m1 m_earth/r^2) / m1 = (G m2 m_earth/r^2) / m2
cancel out values
arrive at the true statement 1=1,
therefore the supposition that a=F/m must have been true.
This doesn't involve measuring the weights experimentally.

russ_watters
Mentor
If you don't need to measure the forces, how did Galileo know to attribute the accelerations to the forces?
He figured out that friction was tripping-up Aristotle. Once you eliminate friction, it is easy to see that force causes acceleration and lack of force means no acceleration.

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Andrew Mason
Homework Helper
But in order to make the proportion $\frac{F_1}{m_1}\textit{=}\frac{F_2}{m_2}$, as mikelepore stated, wouldn't force need to be quantified in some way? If you don't need to measure the forces, how did Galileo know to attribute the accelerations to the forces?
Galileo attributed changes in motion to forces in a general way. But, at least in my understanding, it was not Galileo but Newton who attributed gravitational accelerations to gravitational forces. Galileo determined that in the absence of friction or resistance, all objects fall at the same rate. He determined that the relationship between time, t, of fall and height, h, of fall of an object was $h = at^2/2$. But Galileo did not conclude that they fall at the same rate due to gravitational force being proportional to mass.

AM

Galileo attributed changes in motion to forces in a general way. But, at least in my understanding, it was not Galileo but Newton who attributed gravitational accelerations to gravitational forces. Galileo determined that in the absence of friction or resistance, all objects fall at the same rate. He determined that the relationship between time, t, of fall and height, h, of fall of an object was $h = at^2/2$. But Galileo did not conclude that they fall at the same rate due to gravitational force being proportional to mass.

AM
Ahh so it was Newton who made the connection. But I still don't see how he'd figured a way to have a relative scale of forces. Did Newton use the weights of the bodies or something like spring scales to do this?

Also, based on DH's post Newton seemed to have related the change in momentum to the force, without reference to it's rate change with respect to time. If this is the case, how did we come to interpret it as $\textit{F}\propto{\frac{Δp}{Δt}}$?

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Also, was weight seen as the force due to gravity by Galileo's time? I've read that there was confusion among physicists at the time about the nature of weight; I wasn't sure if it was seen as synonymous with the gravitational force though.

Rap
Why did Newton define force as F=ma instead of stuff like F=m*(a^2) or F=(m^2)*a?
Newton needed to name the quantity ma because it entered into the physical description of things. Since ma corresponds to our intuitive notion of force, he named it force.