in the derivation of F=ma by newton's 2nd law of motion,we consider k as 1?why?

Question

F is force
k is constant of proportionality.
m is the mass
a is acceleration

Answer 1

force entails dimensions (ML^2T-^2).

F= m * a

Force = mass * acceleration or F= m * a

In inertial conditions, when experiments were done to measure the relationship of m, a and F, we did not have the DEFINITION of UNITS-OF-FORCE or word NEWTONS was not assigned to any physical quantity.

It revealed that F makes ‘m’ to accelerate. When ‘m’ increases acceleration decreases. Also, that acceleration varies inversely with the mass. Thus, it was postulated that
F is proportional to “m” and F is proportional to “a”

So F = k (m*a) till then Units of F were not known.

As it was found that 1 kg mass always {{{{{{ACTUALLY}}}}}}} produced 1meter/sec^2 acceleration, therefore, it was decided that let the amount of F be 1 Newton so that k=1. And, constant of proportionality (k) was made dimensionless.

This is very first definition of force 1N = 1 kg-meter/sec^2.

Everywhere else (including that in F=m*g) constant of proportionality (G) has dimensions because now one-NEWTON is defined.

You may not find such aspects of force in any tom-****-harry’s book. Or even if it is written, we tend to jump over it. It needs self-analysis because it is too tiny a fish to spoil the stream (grades in exams).

Answer 2

The derivation stems from impulse, which stems from inertia. By definition...definition...dP = impulse = m dV, when mass is fixed. By experiment we find that dP ~ F dT where F is the average applied force for the period dT.

So it's really dP = kF dT where k is the constant of proportionality. In which case we have dP = m dV = kF dT and kF = m dV/dT = mA. And when we graph A vs F for any given mass m, we find that k = dA/dF = 1 is the slope of the A vs F curve. QED.

Answer 3

a is directly proportional to the force
and indirectly proportional to mass

Thus
force is prop. to ma
but, not equal to ma

therefore we introduce a constant of proportionality,
If f=1,m=1,a=1 unit each,
K=1

F=Kma

Answer 4

In short - we chose the units to make the result come out with K=1. If you take the formula and run it with inconsistant units, you will have to a K equal to something else.