Question

Find the exponent for that makes the following equation dimensionallyconsistent: = 8u/4 x L.The variable is a pressure, = is a flow rate with dimensions of volume per time, hasdimensions of pressure multiplied by time, and both and are lengths

Answer 1

The exponent of L must be 1

Step-by-step 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:

`Pa=((m^3\cdot Pa\cdot s)/(s))/(m^(4))\cdot m^{\text{?}}`

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:

`Pa=(m^3\cdot Pa)/(m^4)\cdot m^{\text{?}}`

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:

`Pa=(Pa)/(m)\cdot m^{\text{?}}`

Now we can see that if L has exponent 1, then the units get cancelled correctly to obtain pressure units, as the equation shows.

`Pa=Pa`