Conceptually, why is Newton's second law of motion $\vec{F} = m\vec{a}$ rather than, say, $\vec{F} = m + \vec{a}$?
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1$\begingroup$ As a side note: The SI units of force, mass and acceleration are N, kg and m/s^2, respectively. Therefore, the second formula does not have the same units on the right and left side. In fact, the sum is not even defined properly, since you are adding kg and m/s^s. $\endgroup$– EthunxxxCommented May 14, 2018 at 9:33
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1$\begingroup$ What conceptually do you think is wrong with $\vec{F}=m\vec{a}$? $\endgroup$– sammy gerbilCommented May 14, 2018 at 9:57
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$\begingroup$ Possible duplicate of why F is to be defined as ma? $\endgroup$– Kyle KanosCommented May 14, 2018 at 10:08
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$\begingroup$ One of the reasons is because then the equation would not be dimensionally consistent. $\endgroup$– Anurag BaundwalCommented May 15, 2018 at 9:17
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3 Answers
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Because we can measure it.
Think of it like this:
- If you push something, it accelerates. Push twice as hard, and you see a doubled acceleration. Push 3 times as hard and you see a trippled acceleration. Acceleration and force clearly follow each other proportionally. This is something we can measure.
But force and acceleration are not equal. From the above we cannot claim $F=a$. They are proportional, not equal.
If you push with 3 N, then the acceleration is maybe 1 $\mathrm{m/s^2}$. The force is 3 times larger. Double the force to 6 N and you will see the acceleration double as well to 2 $\mathrm{m/s^2}$, since they are proportional. This is still a 3 times larger force. We can therefore set up a formula like this, mathematically showing that the force is always 3 times larger than the acceleration: $$F=3a$$
So, what do we call this number "3"? Mathematically it is a proportionality constant. But what is it physically? In some cases it is 3, in other cases another value. It turns out that light objects have a very small value, while heavier objects have larger values. And mathematically, the larger the value is, the smaller is the acceleration caused by the force. So, it is some kind of a "resistance to accelerate".
Let's give it the name mass and the symbol $m$: $$F=ma$$
We also soon realize that there can be many forces, all giving their contribution to the acceleration. We can thus generalize the formula to the sum of all forces acting:
$$\sum F=ma$$
This version of the formula therefore comes from stepwise experimentation. From looking for patterns and dependencies between parameters, such as force and acceleration. It is $\sum F=ma$, and not $\sum F=m+a$ or $\sum F=ma^2$ or $\sum F=m^3/a$ or anything else. If it was, then we would have seen that during the experimentation steps above.
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Newton’s Second Law: The net force on a body is equal to the product of the body’s mass and its acceleration.
It is an experimental law (of direct proportionality):
$$\frac{\overline F}{\overline a}=m_{\text{inertial}}$$
There is no expression like the one you mentioned: $\overline F = \overline m+\overline a$. Furthermore, mass is a scalar and not a vector.
answered May 14, 2018 at 7:41
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Maybe think to find the acceleration. If you think with $$F=m+a$$ then $$a=F-m$$ so the acceleration should be $-m$ if the force is 0. Even, if with math's formulas are reversible: with the same formula, you can have $F$, $m$ or $a$. In physics, I think it is better to think what is the variable ? I apply a force on a mass, then there is an acceleration.
