Problem $37($Shear Stress$).$ A concentric cylinder viscometer may be formed by rotating the inner member of a pair of closely fitting cylinders. The ammiar gap is small so that a linear velocity profile will exist in the liquid sample.

A concentric cylinder viscometer is driven by a falling mass $M$ connected by a cord and pulley to the inner cylinder, as shown in figure $1.$ The liquid to be tested fills the anmiar gap of width a and height $H.$ After a brief starting transient, the mass falls at constant speed $V_m.$

$($a$)$ Assuming that the pulley holding up the mass is frictionless develop an algebraic expression for the viscosity of the liguid in the devier in terms of the known parameters: $M,g,V_m,r,R,a,$ and $H.$

$($b$)$ Use your formula to evaluate the viscosity of the liguid using:

$M=0\mathrm{.}10kg,r=25mm$

$R=50mm,a=0\mathrm{.}20mm$

$H=80mm,V_m=30m\frac{m}{s}$

Answer: $=0\mathrm{.}065\frac{kg}{m*s}.$