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Newton's second Law of Motion states that for a point mass, $\vec F = m \vec a$. This is a law and not a definition. So, this law only makes sense if all the physical quantities appearing in this law are pre-defined.

$\vec a$ is pre-defined because $\vec a =\frac {d \vec v}{dt}=\frac{d^2 \vec r}{dt^2}$, where $\vec v$ and $\vec r$ represent velocity vector and position vector respectively. $m$ is the mass of the concerned point mass and is one of the fundamental physical quantities so it is definitely pre-defined.

But what about force ? How do we define it quantitatively ? The only definition which I remember says that one newton is the amount of force required to accelerate $1$ $kg$ of body by $1$ $m/s^2$. But that is just defining force with $\vec F = m \vec a$, which is just circular reasoning.

So, what is "force" really ?? I mean ...... on a fundamental level ..... without using any kind of circular reasoning ..... how do we actually define the term "force" quantitatively???? I mean, surely there must be some way to define force, right ?

EDIT : Is it even possible to define force ?

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    $\begingroup$ Possible duplicate of Are Newton's "laws" of motion laws or definitions of force and mass? $\endgroup$ – Emilio Pisanty Sep 22 at 17:40
  • $\begingroup$ @Quadro You said “This is a law and not a definition” I disagree that laws cannot be definitions. I would consider Newton’s 2nd law to be a definition of force $\endgroup$ – Dale Sep 22 at 18:11
  • $\begingroup$ we can measure mass and acceleration, so they have a measurable quantity, since force equals mass times acceleration, force has a measurable quantity, a newton $\endgroup$ – Adrian Howard Sep 22 at 18:20
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We don’t define “force” in general; we define specific forces as specific functions of position and velocity. For example, Newtonian gravitational force is $\mathbf{F}=-\frac{GMm}{r^2}\hat{\mathbf{r}}$ and Lorentz force is $\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B})$. Putting such forces into Newton’s Second Law then produces a differential equation of motion predicting how things move. There is no circularity when you think of force in this way.

Without a specific force law, force is simply “any interaction which makes an object accelerate”, which has little predictive usefulness. But with specific force laws, you can understand in detail and sometimes to extreme precision how the physical world works.

By following this approach, we have discovered that all phenomena that we observe are explainable in terms of only four fundamental forces: gravity, electromagnetism, the weak nuclear force, and the strong nuclear force. This is one of the greatest intellectual achievements of humankind.

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  • $\begingroup$ How can a law based on force make sense if we do not know what force really is ? By the way, I did not downvote. $\endgroup$ – Quadro Sep 22 at 17:41
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    $\begingroup$ It makes sense because physics is about constructing mathematical models of reality that explain observation, not “knowing” what things “are”. We explain by calculating and predicting! $\endgroup$ – G. Smith Sep 22 at 17:45
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    $\begingroup$ Thinking that $F=ma$ defines force is rather like thinking that $E=mc^2$ defines energy. Don’t do it. Both are relationships, not definitions. $\endgroup$ – G. Smith Sep 22 at 18:19

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