I don't understand how $F$ equals $ma$

Question

I am a beginner in Physics and currently doing laws of motion. So, I don't really understand how the second law is true. It's just that I don't understand how it makes sense.

So, we start with the assumption that a and m are directly proportional to F. Now, how does it follow from it that F=ma where k=1 is the constant of proportionality ? All that proportionality implies is that with increase in a or m or both, the force as well, increases or vice versa.

Answer 1

Mathematically Directly proportional means if you increase acceleration twice force will also increase two times and similarly for mass.

So assuming mass m to be constant, $$F \propto a$$ Similarly, assuming acceleration to be constant $$F \propto m$$ Now if you increase a x times and m y times then by above proportionality F will increase by $x.y$ times. For understanding this you can assume the process to happen in steps, e.g. First increasing the mass and then increasing the acceleration.

So $$F \propto ma$$ $$F =k .ma$$ Now in SI unit system, unit of force is defined by taking k=1, I.e. 1 N forces is standardised to be force needed for moving 1 kg mass with $1 ms^{-1}$ acceleration So k=1

Answer 2

So, we start with the assumption that a and m are directly proportional to F.

That is not true. Besides not being an assumption, the statement should have been:

Acceleration $a$ is proportional to net force $F$ with the coefficient of proportionality being the mass $m$.

I don't see the point to have two constants of proportionality between $F$ and $a$ is in $F = (k m) a$ as you can roll it into one. Experimentally it has been proven that the proportionality mass $m$ (called internal mass) is identical to the gravitational mass. See Eötvös experiment.

Answer 3

Think of it this way. You can do various things to an object to make it change its velocity. For example, you can push it, pull on it, put a magnet by it (if it is magnetic), drop it and so on. We give the effect of all those methods a common label, which is 'force'.

If you apply a force to one body, and then apply the same force to two identical bodies fastened together, the rate of change in velocity that the force brings about in the second case is half the rate that it brings about in the first. Through experiments of that sort you can find that the more mass you apply a force to, the smaller the effect it has in terms of the rate of change of velocity.

You can do another set of experiments in which you apply different forces to the same object. You will find from those that the more force you apply to an object, the greater is the effect on the rate of change of its velocity.

Experimentally, therefore, you can show that there is a directly proportional relationship between the force you apply to a given mass and the rate of change in velocity that results from the application of the force. If you define your units sensibly, you can make the constant of proportion one.

So, practically what Newton's Second Law means is that if you subject a range of different bodies to a range of different forces, in each case the resulting acceleration will be related to the size of the force and the mass of the object by the formula F=ma.

Answer 4

Newton’s second law says that acceleration is proportional to Force. The proportionality constant is the mass. In a Newtonian context this is the definition of mass.

Answer 5

Let F=mak, redefine a new quantity F'=F/k. There, now you have F'=ma.

These ideas are mutually defined. Yes, in a metaphysical sense all we observe is some proportionality but when we sit down to define quantities, we can make such convenient choices. In fact, in more advanced physics, this is the idea of setting physical constants like c=1

Answer 6

Essentially speaking, Newton's 2nd law states that the force applied by a body is equal to the rate of change of its momentum.

Let's say the initial momentum of a body was $mv$, m being its mass and v its velocity, and finally it comes down to $mu$, where u is the reduced (assume) velocity.

The change in momentum is obviously $mv-mu = m(v-u)$.

We assume this change in momentum takes place over a time t. By definition of 'rate', the rate of change of the momentum is then $\frac{mv-mu}{t}$ = $\frac{m(v-u)}{t}$.

Acceleration is defined as the rate of change of velocity, or in other words:

$\frac{v-u}{t} = a$

Plugging this in, we get the rate of change of momentum to be $ma$, which according to Newton's definition is equal to the force. Mathematically:

$F = \frac{m(v-u)}{t} = ma$

To look at it another way, Newton found that:

$F \propto a$

$F \propto m$

thus:

$F \propto ma$

To remove the proportionality sign, we needed a constant of proportionality, which, to define 1N as 1kg times 1m/s$^2$, Newton took to be 1, giving us:

$F = ma$

Answer 7

@rayhan, So basically they say that Force is the derivative of momentum w.r.t. time (im sorry if you haven't yet studied basic calculus and cheers if you have),

So F=dP/dt,

and we can write it like , F= d(m . v)/dt

so by applying the product rule we get, F= [ dm/dt * v ]+[ dv/dt * m]

now if the mass of the body is constant (it only works for constant masses) the dm/dt term = 0,

so we get F = dv/dt * m

where dv/dt= acceleration,

so F = ma

and there you have it, f=ma

sorry if it's hard to read or to understand, thanks.