The force $F$ on a sphere of radius ‘$a$’ moving in a medium with velocity ‘$v$’ is given by $F = 6\pi\eta av$

Question

The force $F$ on a sphere of radius ‘$a$’ moving in a medium with velocity ‘$v$’ is given by $F = 6\pi\eta av$. The dimensions of $\eta$ are:

Accepted Answer

To find the dimensions of the coefficient of viscosity ($\eta$), we use the provided formula for viscous force (Stokes' Law): $F = 6\pi\eta av$. Rearranging the formula for $\eta$, we get $\eta = F / (6\pi av)$. According to the sources, the dimensional formulae for the relevant quantities are:

1.  **Force ($F$):** $[MLT^{-2}]$ .
2.  **Radius ($a$):** As a measure of length, its dimension is $[L]$ .
3.  **Velocity ($v$):** $[LT^{-1}]$ .
4.  **$6\pi$:** This is a numerical constant and is dimensionless .

Substituting these into the equation for $\eta$:
$[\eta] = \frac{[MLT^{-2}]}{[L][LT^{-1}]} = \frac{[MLT^{-2}]}{[L^{2}T^{-1}]} = [ML^{-1}T^{-1}]$.

This dimensional formula corresponds to Option A .

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