# What is the true quantitative definition of “force”? [duplicate]

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 ?

• Possible duplicate of Are Newton's "laws" of motion laws or definitions of force and mass? – Emilio Pisanty Sep 22 at 17:40
• @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 – Dale Sep 22 at 18:11
• 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 – Adrian Howard Sep 22 at 18:20

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.
• 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. – G. Smith Sep 22 at 18:19