Answer to Question #227771 in Classical Mechanics for J17

Question

A drain has a rectangular profile. W is the width of the drain in meters, and

D is the depth of the water in meters.

In the case that the depth is much less than the width the amount of water that flows per second through a rectangular drain is given by the following formula:

Q = (A/n)*(D^2/3)*(S^1/2) ------------- (1)

In which:

- Q is the volume of water that

flows per second through the

drain (in m3/s);

- A = W×D, the cross-sectional

area of the drain up to the level

of the water (in m2)

- n is a parameter describing the

resistance to the flow of water;

- S is the gradient of the river

(in m/m).

Use formula 1 to derive the units of n.

Answer 1

$$[Q]=\frac{[A]}{[n]}[D]^{\frac{2}{3}}[S]^{\frac{1}{2}}\\ L^3T^{-1}=\frac{L^2}{[n]}L^{\frac{2}{3}}T^{\frac{1}{2}}\\ [n]=L^{-\frac{1}{3}}T^{-\frac{3}{2}}$$