RESNICK HALLIDAY QUESTION DOUBT

subho bhayiya you didn't understand my doubt.......there's no problem in solving the question whatever no of dependent quantities are given(whether mass or g or any other such thing) ......the doubt is - how can you assume 'k' to be dimensionless until it is said that in the qustion

suppose a is proportional to m^x , rⁿ and v^m

thus a is proportional to (m^xrⁿv^m)
or, a=k.(m^xrⁿv^m) where k is a dimensionless constant...

now apply dimensional analysis

this is your solution but how can you assume 'k' to be dimensionless(we all know by studying circular motion that it is dimensionless but we can't apply that knowledge here)

well u hav to assume it to be dimensionless otherwise the dimensional formula can itself go wrong!!!!

no convincing answers till now
@akhil
the dimensional formula won't go wrong but the actual formula will go wrong...just for example let us assume 'k' has dimensions of time
k = [T] then the formula will become(by solving using dimensional analysis):

a = k.v³r² (whereas you know that actual formula is a = k.v²r )

dimensionally the formula is still correct but this is not the actual formula .
i remember one line regarding this given in many books:

A DIMENSIONALLY CORRECT EQUATION NEED NOT BE ACTUALLY CORRECT BUT A DIMENSIONALLY WRONG EQUATION MUST BE WRONG ...

so this is my only doubt that even when we know that all physical constants are not dimensionless(example is universal gravitational constant) ,still why do we assume these constants to be dimensionless while solving such problems

yes subhomoy i got a hint that you wanted to say this when you wrote the line
look consider the different cases in which accn might depend..in your last to last post in this thread and trust me, initially i was really convinced
but when I see the inverse square equations:

1. F= Gm₁m₂r² (universal gravitation law)

2. F = kqr²

and other inverse square equations
i sense a big difference in what you say and what the reality is
just taking example of 1.(ie universal gravitation law)
in this equation also m₁, m₂ and 'r' are the only terms that 'F' depends upon but still the constant is not dimensionless
if you try formulate any of inverse square laws using dimensional analysis like in the topmost question you'll surely get all the formulas wrong
now there can be only two possibilities:
1. either all the inverse square laws are a big exception to dimensional analysis
or
2. there is a very good and convincing reason behind all this