Question

Obtain stock Law dimesionally

Answer 1

Answer:

First we must ask: what can this drag force depend on?

Obviously, it depends on the size of the ball: let’s say the radius is a, having dimension L.

It must depend on the speed v, which has dimension LT−1.

Finally, it depends on the coefficient of viscosity η which has dimensions ML−1T−1.

The drag force F has dimensions [F]=MLT−2: : what combination of [a]=L,  [v]=LT−1 and [η]=ML−1T−1 will give [F]=MLT−2 ?

It’s easy to see immediately that F must depend linearly on η, , that’s the only way to balance the M term.

Now let’s look at F/η, which can only depend on a and v. [F/η]=L2T−1. The only possible way to get a function of a,v having dimension L2T−1 is to take the product av.

So, the dimensional analysis establishes that the drag force is given by:

F=Caηv

where C is a constant that cannot be determined by dimensional considerations.