ANSWER
The exponent of L must be 1
EXPLANATION
We have to write the dimensions of each variable in the equation. 8 and π are constants and have no units. The units of the variables, in the international system of units, are:
• Q = m³/s
,
• u = Pa·s
,
• r = m
,
• L = m
,
• P = Pa
Replace these into the equation. For this problem we can ignore the constants:
[tex]Pa=\frac{\frac{m^3\cdot Pa\cdot s}{s}}{m^{4}}\cdot m^{\text{?}}[/tex]
We want to have Pascal units on both sides for this to be dimensionally consistent. Note that the seconds in the numerator's fraction get cancelled:
[tex]Pa=\frac{m^3\cdot Pa}{m^4}\cdot m^{\text{?}}[/tex]
To have Pascal on both sides we need the same exponent in the meters. If we cancel m³ with m⁴, we have m¹ in the denominator:
[tex]Pa=\frac{Pa}{m}\cdot m^{\text{?}}[/tex]
Now we can see that if L has exponent 1, then the units get cancelled correctly to obtain pressure units, as the equation shows.
[tex]Pa=Pa[/tex]
