Sample Problem

Air Resistance 1: Dimensional analysis

We know that as an object passes through the air, the air exerts a resistive force on it. We can get some idea as to the form of this force by dimensional analysis. Suppose we have a spherical object of radius R and mass m. What might the force plausibly depend on?

It might depend on the properties of the object. The only ones that seem relevant are m and R.
It might depend upon the object's coordinate and its derivatives: x, v, a, ....
It might depend on the properties of the air, such as the density, r.

(a) Explain why it is plausible that the force the air exerts on a sphere depends on R but implausible that it depends on m.

(b) Explain why it is plausible that the force the air exerts depends on the objects speed through it, v, but not on its position, x, or acceleration, a.

(c) Using dimensional analysis, construct a plausible form for the force that air exerts on a spherical body moving through it.

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Page last modified October 15, 2002: D28