Formula of forces

1. Jun 16, 2016

foo9008

1. The problem statement, all variables and given/known data
from the notes , the author stated that force has the formula of ρ(L^2)(v^2) and also = ρ(L^3)
i think there's something wrong with the ρ(L^3)

2. Relevant equations

3. The attempt at a solution
IMO , ρ(L^2)(v^2) can also be written as ρ(L^4)(T^2) , so Force is proportional to L^4 , am i right . ? IMO, the prototype force should be [ (100^4) ] x 0.12 N

correct me if i am wrong ...

2. Jun 16, 2016

tommyxu3

I'm not sure, but in your picture, one is $\rho$ and the other is $\rho_r.$ Maybe there is detailed definitions in the problem?

3. Jun 16, 2016

foo9008

rho r is actually r , it's the pi buckingham theorem

4. Jun 17, 2016

haruspex

I guess you mean ρ(L^4)(T^-2), but either way it does not follow. The whole point of figuring out these dimensionless expressions is that you cannot change the scale of L and assume T fixed, etc. There are physical constants in the system, like viscosity, which impose a relationship between the fundamental dimensions. Changing L keeping T etc. fixed will therefore change the viscosity.
Similarly, density fixes a relationship between M and L.

5. Jun 17, 2016

foo9008

Then, what is the correct formula of force in this question?

6. Jun 17, 2016

foo9008

so , the author is correct ? it is (rho)(L^2)(L ) , where (v^2) = L ??

7. Jun 17, 2016

haruspex

I'm hampered by not knowing what Fr stands for in the first line.
I presume F=ρL2V2 comes from some earlier work.
If we accept both of those equations, the rest follows.

8. Jun 17, 2016

foo9008

So, the authors working is correct??

9. Jun 17, 2016

haruspex

I'm not sure. I don't know where the first line of equations comes from, or what Fr represents. I'm surprised to see any reference to g here. How is gravity relevant? If gravity were to increase but the densities, masses, and lengths stay the same, those equations seem to say the velocity would increase. Why?