## 2

The number of dimensionless groups that will be obtained is equal to the number of variables less the minimum number of fundamental dimensions involved in these variables (7-3=4 groups in this problem).

Step 3: Choose a set of reference variables. The choice of variables is arbitrary, except that the following criteria must be satisfied:

1. The number of reference variables must be equal to the minimum number of fundamental dimensions in the problem (in this case, three).

2. No two reference variables should have exactly the same dimensions.

3. All the dimensions that appear in the problem variables must also appear somewhere in the dimensions of the reference variables.

In general, the procedure is easiest if the reference variables chosen have the simplest combination of dimensions, consistent with the preceding criteria. In this problem we have three dimensions (M, L, t), so we need three reference variables. The variables D, e, and L all have the dimension of length, so we can choose only one of these. We will choose D (arbitrarily) as one reference variable:

The dimension t appears in V, AP, and but V has the simplest combination of dimensions, so we choose it as our second reference variable:

We also need a reference variable containing the dimension M, which could be either p or Since p has the simplest dimensions, we choose it for the third reference variable:

Our three reference variables are therefore D, V, and p.

Step 4: Solve the foregoing ''dimensional equations'' for the dimensions (L, t, M) in terms of the reference variables (D, V, q), i.e.,

Step 5: Write the dimensional equations for each of the remaining variables. Then substitute the results of step 4 for the dimensions in terms of the reference variables:

Step 6: These equations are each a dimensional identity, so dividing one side by the other results in one dimensionless group from each equation:

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