# Defining Force

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#1
Issac Newton not only introduced the concepts of velocity and acceleration, which can be measured by measuring distances and time, but also introduced the concepts of mass and force. So, technically, is the oldest equation in Physics. However, putting the definition of mass across, the other day a friend asked how did Newton know that exactly , and why did he not just define the force to be . So I have made this thread to see what others think.
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2 years ago
#2
(Original post by Absent Agent)
Issac Newton not only introduced the concepts of velocity and acceleration, which can be measured by measuring distances and time, but also introduced the concepts of mass and force. So, technically, is the oldest equation in Physics. However, putting the definition of mass across, the other day a friend asked how did Newton know that exactly , and why did he not just define the force to be . So I have made this thread to see what others think.

Isaac Newton did not come out with .

The modern F=ma form of Newton's second law nowhere occurs in any edition of the Principia even though he had seen his second law formulated in this way in print during the interval between the second and third editions in Jacob Hermann's Phoronomia of 1716. Instead, it has the following formulation in all three editions: A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed. In the body of the Principia this law is applied both to discrete cases, in which an instantaneous impulse such as from impact is effecting the change in motion, and to continuously acting cases, such as the change in motion in the continuous deceleration of a body moving in a resisting medium. Newton thus appears to have intended his second law to be neutral between discrete forces (that is, what we now call impulses) and continuous forces. (His stating the law in terms of proportions rather than equality bypasses what seems to us an inconsistency of units in treating the law as neutral between these two.)

The above paragraph is taken from https://plato.stanford.edu/entries/n...pia/#NewLawMot

If I don’t remember wrongly, F = ma is from Euler.

https://en.wikipedia.org/wiki/Euler%27s_laws_of_motion

Hope it helps.
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#3
(Original post by Eimmanuel)
Isaac Newton did not come out with .

The modern F=ma form of Newton's second law nowhere occurs in any edition of the Principia even though he had seen his second law formulated in this way in print during the interval between the second and third editions in Jacob Hermann's Phoronomia of 1716. Instead, it has the following formulation in all three editions: A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed. In the body of the Principia this law is applied both to discrete cases, in which an instantaneous impulse such as from impact is effecting the change in motion, and to continuously acting cases, such as the change in motion in the continuous deceleration of a body moving in a resisting medium. Newton thus appears to have intended his second law to be neutral between discrete forces (that is, what we now call impulses) and continuous forces. (His stating the law in terms of proportions rather than equality bypasses what seems to us an inconsistency of units in treating the law as neutral between these two.)

The above paragraph is taken from https://plato.stanford.edu/entries/n...pia/#NewLawMot

If I don’t remember wrongly, F = ma is from Euler.

https://en.wikipedia.org/wiki/Euler%27s_laws_of_motion

Hope it helps.
Thank you for your response! I always took for granted that F=ma was credited to Newton, but it's also interesting that what he said is his Principia as a law of motion i assume could be written as , but Euler seems to have derived F=ma from first principles.
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2 years ago
#4
(Original post by Absent Agent)
Issac Newton not only introduced the concepts of velocity and acceleration, which can be measured by measuring distances and time, but also introduced the concepts of mass and force. So, technically, is the oldest equation in Physics. However, putting the definition of mass across, the other day a friend asked how did Newton know that exactly , and why did he not just define the force to be . So I have made this thread to see what others think.
Actually,for constant mass,accelaration is always exactly in direct relation with the force applied.Thus,Newton took F=ma instead of F=m+a.
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2 years ago
#5
(Original post by Absent Agent)
Thank you for your response! I always took for granted that F=ma was credited to Newton, but it's also interesting that what he said is his Principia as a law of motion i assume could be written as , ...
I had not read Principia, so I don't really know if the assumption is valid.

To my limited knowledge, I am not sure did Newton "define" motion in Principia and it seems that Newton never presented the “three laws of motion” in the form of equations.

(Original post by Absent Agent)
… but Euler seems to have derived F=ma from first principles.
Again, I am not sure about this. If you can support it with reference(s), I would be glad. I would really like to know how did Euler arrive F =
ma. Thanks.
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#6
Actually,for constant mass,accelaration is always exactly in direct relation with the force applied.Thus,Newton took F=ma instead of F=m+a.
Well, in both cases, acceleration is directly proportional to the force.

(Original post by Eimmanuel)
I had not read Principia, so I don't really know if the assumption is valid.

To my limited knowledge, I am not sure did Newton "define" motion in Principia and it seems that Newton never presented the “three laws of motion” in the form of equations.
That's true! There are English translations of the Principia, and it seems that his laws of motion are stated as propositions rather than mathematical expression, but apparently, he provided geometrical proofs for the inverse square law of gravity. Also, there are also some definitions stated before the laws, and the one below is one of them:

"The motion of a body is the sum of the motions of the parts; and therefore in a body with double quantity, with equal velocity, the motion is double; with twice the velocity, it is quadruple."

I have difficulty understanding whether that definition has got anything to with his second law, but it seems that taking F=m+a assumes that a mass with it's acceleration would not be "conjunct" quantities. As a result, according to F=m+a, a body of a certain mass with zero acceleration would have require force for its state of motion (F=m+0).

Attachment 709102

In fact, Newton seems to refer to motion as the quantities of mass and velocity, but I don't know if he also means the product of mass and velocity.

(Original post by Eimmanuel)
Again, I am not sure about this. If you can support it with reference(s), I would be glad. I would really like to know how did Euler arrive F =
ma. Thanks.
Well, I thought the link to Wikipedia you provided in your first post also contained derivations of Euler's laws, but I really cannot make sense of them.
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2 years ago
#7
Actually Newton wrote:
F = dp/dt
Given p = mv and a = dv/dt
--> F = d (mv)/dt
F = m dv/dt
F = ma
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2 years ago
#8
(Original post by williamnguyen)
Actually Newton wrote:
F = dp/dt
Given p = mv and a = dv/dt
--> F = d (mv)/dt
F = m dv/dt
F = ma

Note that F = ma is not from Newton but from Euler. This is a fact. And Newton never presented the “three laws of motion” in the form of equations.
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2 years ago
#9
(Original post by Absent Agent)
Well, in both cases, acceleration is directly proportional to the force.....

Not really. When we say one quantity y is proportional to another quantity x, it means y = kx. When we write y = kx + c, it means y varies linearly with respect to x.

(Original post by Absent Agent)
... Well, I thought the link to Wikipedia you provided in your first post also contained derivations of Euler's laws, ...

It contains the derivation but how did Euler derive F = ma is another story. Did Euler extend the work of Newton or did Euler derive F = ma independently?

(Original post by Absent Agent)
... but I really cannot make sense of them.

It depends on what level of physics you are doing. The derivation is for rigid bodies instead of point particle. You need vector analysis and vector calculus to appreciate the maths.
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