assuming that the largest mass that can be moved by a flowing river depends on the velocity of flow,density of river water,and acceleration due to gravity,show that the mass varies as the sixth power of velocity of flow.
We posit that M = C v^a * p^b * g^c with C a dimensionless constant, and a , b and c (dimensionless) exponents. Let us determine what values of a, b and c are allowed to make the formula dimensionally correct.
v is s speed, so length/time ( L / T) . p is a density, so mass/volume (m / L^3) g is an acceleration, so length / time^2 ( L / T^2)
This means that in terms of dimensions, the equation is m = (L / T)^a * (m / L^3)^b * ( L / T^2)^c So, combining powers in each quantity (L, m, T):
m = L^(a - 3b + c) * m^b * T^(-a -2c)
Whence we conclude that
b = 1 a - 3b + c = 0 -a -2c = 0
b = 1 a + c = 3 a + 2 c = 0
The latter two equations are solved by c = -3 a = 6