Consider a thin square plate floating on a viscous liquid in a large tank. The height $h$ of the liquid in the tank is much less than the width of the tank. The floating plate is pulled horizontally with a constant velocity $u_0$. Which of the following statements is (are) true?
$(A)$ The resistive force of liquid on the plate is inversely proportional to $h$
$(B)$ The resistive force of liquid on the plate is independent of the area of the plate
$(C)$ The tangential (shear) stress on the floor of the tank increases with $u _0$
$(D)$ The tangential (shear) stress on the plate varies linearly with the viscosity $\eta$ of the liquid
IIT 2018, Medium
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Viscous force $F =-\eta A \frac{ dv }{ dh }$
$F =\eta A \left(\frac{u_0}{ h }\right)$
$\Rightarrow F \propto \frac{1}{ h }$
Shear stress $=\frac{ F }{ A } \propto u _0$
$\frac{ F }{ A } \propto \eta$
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