Question

Determine that both sides of the above equation are dimensionally consistent m

Answer 1

Step 1. Find the dimensions of the left-hand side of the equation.

In this part of the equation what we have is:

`(\Delta P)/(L)`

Here Δ represents a change in P, in the end, ΔP and P have the same units

And we are indicated that the units of P and L are:

`\begin{gathered} P=\lbrack N\cdot m^2\rbrack \\ L=\lbrack m\rbrack \end{gathered}`

Note: we use [ ] to represent that they are units.

Thus in this left part of the equation, the units are:

`(\lbrack N\cdot m^2\rbrack)/(\lbrack m\rbrack)`

Simplifying the m:

`\lbrack N\cdot m\rbrack`

Step 2. We have found that the units on the left-hand side are N*m

Now, we have to find the units on the right-hand side.

On the right-hand side of the equation we have:

`(300(1-\phi^2)\mu U)/(2D^2\phi^3)+(175\rho U^2)/(100D\phi^3)`

The units of the variables are:

Substituting the units (we can ignore the numbers and the dimensionless terms):

`(300(1-\phi^2)\mu U)/(2D^2\phi^3)+(175\rho U^2)/(100D\phi^3)=(\lbrack Pa\cdot s\rbrack\lbrack(m)/(s)\rbrack)/(\lbrack m^2\rbrack)+(\lbrack(kg)/(m^3)\rbrack\lbrack(m^2)/(s^2)\rbrack)/(\lbrack m\rbrack)`

Simplifying the divisions and multiplications between the units:

`(\lbrack Pa\cdot m\rbrack)/(\lbrack m^2\rbrack)+(\lbrack(kg)/(ms^2)\rbrack)/(\lbrack m\rbrack)`

Simplifying further:

`\lbrack(Pa)/(m)\rbrack+\lbrack(kg)/(m^2s^2)\rbrack`

Since the units of Pascals can be also represented as:

`Pa=\lbrack(kg)/(m\cdot s^2)\rbrack`

The second term can also be expressed as Pa/m:

`\lbrack(Pa)/(m)\rbrack+\lbrack(Pa)/(m)\rbrack`

The addition of two terms with the same units does not change the units, thus, the units on the right-hand side are:

`\lbrack(Pa)/(m)\rbrack`

Step 3. Compare.

The units on the left-hand side are:

`\lbrack N\cdot m\rbrack`

And the units on the right-hand side are:

`\lbrack(Pa)/(m)\rbrack`

To compare them, let's convert the units of the left-hand side to Pascals using the following known relation between Newtons and Pascals:

`\lbrack N\rbrack=\lbrack(Pa)/(m^2)\rbrack`

Using this, the units of the left-hand side are:

`\lbrack N\cdot m\rbrack=\lbrack(Pa)/(m^2)\cdot m\rbrack=\lbrack(Pa)/(m)\rbrack`

As you can see, The units of both sides are Pa/m, thus we have proven that the two sides are dimensionally consistent.