Newton's second law of motion -- Why is it F=m*a and not F=m+a?

[akerkarprashant]

TL;DR Summary

F=m*a equation

Why F=m*a i.e product or multiplication and not F=m+a? i.e addition or summation?

 

[PeroK]

Summary:: F=m*a equation

Why F=m*a i.e product and not F=m+a? i.e addition?

Because Newton's first law says that $a = 0$, when $F = 0$.

 

[Halc]

Primarily, because F=m+a doesn't predict what we empirically measure.

Mathematically, m and a are different units, one each a scalar and a vector, so it is mathematically meaningless to add them.

 

[kuruman]

Summary:: F=m*a equation

Why F=m*a i.e product or multiplication and not F=m+a? i.e addition or summation?

Because it predicts that a 9.81 kg mass would float in midair if released at a point near the Earth's surface where the acceleration of gravity is -9.81 m/s². Don't try this at home.

 

[akerkarprashant]

Thanks.

So we can add Scalar + Scalar, Vector + Vector but not Scalar + Vector quantity?
While Scalar * Scalar, Vector * Vector and Scalar * Vector quantity is possible?

 

[PeroK]

Thanks.

So we can add Scalar + Scalar, Vector + Vector but not Scalar + Vector quantity?
While Scalar * Scalar, Vector * Vector and Scalar * Vector quantity is possible?

You can mutiply pretty much any two quantities. If they are both vectors you need to use the dot product or the cross product.

You can only add quantities that have the same units. E.g. if both have units of mass; or, if both have units of mass times length divided by time squared. Etc.

 

[Ibix]

You can only add quantities that have the same units. E.g. if both have units of mass; or, if both have units of mass times length divided by time squared. Etc.

Indeed. What is one second more than one kilogram? It doesn't make sense. But one second times one meter per second gives you one meter, the distance traveled in the time.

 

[akerkarprashant]

Thanks.

https://mathvault.ca/hub/higher-math/math-symbols/common-math-symbols/

https://mathvault.ca/hub/higher-math/math-symbols/algebra-symbols/

We always check why Sir Isaac Newton has written
the equation F=m*a in his Second law of Motion and
why not the equations could have been
F=m+a
F=m-a
F=a-m
F=a^m
F=m^a
F=m/a
F=a/m
F=m%a
F=a%m
F=π*m*a where π=22/7
F=m√a
F=a√m
F=m+ai - complex
F=a+mi - complex
The unit of Force is Newton in MKS.
The unit of Force is Dynes in CGS.

 

[jbriggs444]

[...]
F=m^a
F=m/a
[...]

Is this the infinite monkey theorem applied to theoretical physics?

 

[italicus]

The main reason why F=ma and not m+a, further to dimensional nonsense, is that a material particle of mass m remains at rest wrt to your reference frame , supposed inertial, if it was initially at rest and no force F acts on it. But not only: because a = dv/dt is the derivative of velocity wrt time, in the case of v = const this derivative is null. So the particle can keep constant velocity, and no force F is required for that.
Quantities F, v, a , are vectors.

In conclusion: a material particle remains at rest or in rectilinear uniform motion, in an inertial reference frame, until a force F changes this state, causing an acceleration a= dv/dt.

More correctly, the second law of dynamics should be written:

F = dp/dt

Where p = mv is another vector quantity, the particle momentum.

 

[kuruman]

Is this the infinite monkey theorem applied to theoretical physics?

It reminds me of the old Sidney Harris cartoon $\dots$

 

[Delta2]

As others have essentially said , that's the law of nature that $\vec{F}=m\cdot \vec{a}$ (to be more precise Newton state it as $\vec{F}=\frac{d\vec{P}}{dt}$). The laws of nature could 've been different, for example Maxwell's equations could ve been different and Newtons Laws could ve been different, it could 've been $F=ma^2$ or $F=ma^n$ for some $n\in \mathbb{N}$, but God (moderators sorry if I mention God here I hope you don't mind) or the universe by it self selected $F=ma$ .

You can write simulations, that simulate an alternate Universe in some programming language, where we have alternate forms of the law for example $F=ma^2$ and see how things behave. Try to solve the harmonic motion (which won't be harmonic anymore) for $m(\frac{d^2 x}{dt^2})^2=-kx$. I can tell you that the universe would be slower if for example $F=ma^2$.

 

[italicus]

It seems so simple to say , but we have to deeply thank Galileo, Newton and others. Let me make a short digression, forgive me.
For almost two thousand years the scientific world was pervaded by the Aristotle ‘s physics, now we know it was wrong. He spoke of four elements, Earth water air and fire , and distinguished “natural motion “ from “violent motion “, But overall he was unable to explain why a stone, thrown in the air, continued to move without any apparent cause. He had no idea of the principle of inertia. Have a look at this short lesson:

https://aether.lbl.gov/www/classes/p10/aristotle-physics.html

Very often, there is progress in science not only when we learn something new, but also when we are able to look at known facts from a different point of view.

 

[jbriggs444]

Why F=m*a

Because force is proportional to mass: If you have twice as much mass, you have to push twice as hard to accelerate it at the same rate.

Because force is proportional to acceleration: If you want twice as much acceleration, you have to push twice as hard.

Proportionality is the important bit. It turns out that if one thing is proportional to two others then it is proportional to the product of those two things.

For example, area is proportional to length. Area is proportional to width. Area is proportional to the product of length and width.

Next lesson: The constant of proportionality, $F=kma$ and coherent units.

 

[vanhees71]

The answer is that Newton was after figuring out how nature works and not after writing down mathematical expressions which don't make any sense!

 

[berkeman]

The answer is that Newton was after figuring out how nature works and not after writing down mathematical expressions which don't make any sense!

With that, we will close this thread. Thank you everybody for trying to help the OP.