Question

When a small sphere moves at a low speed through a fluid, the viscous force F opposing the motion is found experimentally to depend on the radius r, velocity v, and the coefficient of viscosity n of the fluid. Find the force dimension of the force F

Answer 1

$$F=r^av^b\eta^c$$

$$\dim [F]=(\dim [r])^a(\dim [v])^b(\dim [\eta])^c$$

$$MLT^{-2}=(L)^a(LT^{-1})^b(ML^{-1}T^{-1})^c$$

$$c=1$$

$$-2=-b-1\to b=1$$

$$1=a+1-1\to a=1$$

Thus,

$$F=\eta rv$$