Question

The viscous charge F between two layers of liquid with specific area of contact A in a region of velocity gradient dv/dt is given by F = integration sign A dv ÷dt where the constant is the cofficient of the viscosity .what is the dimension of the constant.

Answer 1

The Newton's law for viscosity

$$F = \eta A{{\partial v} \over {\partial x}}$$

where $F$ - force, has the dimension $[{\rm{LM}}{{\rm{T}}^{ - 2}}]$, $A$ - area of contact, hast the dimension $[{{\rm{L}}^2}]$ and ${{\partial v} \over {\partial x}}$ - velocity gradient has the dimension ${\rm{[}}{{\rm{T}}^{ - 1}}]$ (because $v$ has the dimension $[{\rm{L}}{{\rm{T}}^{ - 1}}]$ and $x$ has the dimension $[{\rm{L}}]$ ). Let $\eta$ has the dimension $[X]$. Then, using the Newton's law for viscosity

$$[{\rm{LM}}{{\rm{T}}^{ - 2}}] = [X][{{\rm{L}}^2}][{{\rm{T}}^{ - 1}}]$$

Thus

$$[X] = [{\rm{M}}{{\rm{L}}^{ - 1}}{{\rm{T}}^{ - 1}}]$$