60.  The volume of liquid flowing per second is called the volume flow rate Q and has the dimensions of [L]3/[T]. The flow rate of a liquid through a hypodermic needle during an injection can be estimated with the following equation:

The length and radius of the needle are L and R, respectively, both of which have the dimension [L]. The pressures at opposite ends of the needle are P2 and P1, both of which have the dimensions of [M]/{[L][T]2}. The symbol h represents the viscosity of the liquid and has the dimensions of [M]/{[L][T]}. The symbol p stands for pi and, like the number 8 and the exponent n, has no dimensions. Using dimensional analysis, determine the value of n in the expression for Q.

 

We now put the units in for each of the terms

 

Since the Pressures are subtracting and they have the same units we can simplify this set of units to the following

Now cancel units where possible and we get

 

So the units come down to

The units work if

So

 

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This page last updated on January 11, 2020