Question

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 η of the liquid.

Answer 1

Given that the liquid is viscous, it means there will be presence of viscous force in it. We remove the plate with a constant horizontal velocity ${{u}_{0}}$, so we can use the concept of viscous force to arrive at the solution.

Complete step by step answer:
Viscous force is given by the formula,
$F=-\eta A\dfrac{\Delta v}{\Delta y}$
Here, $\eta$ is the coefficient of viscosity and A is the area of the contact. Here minus sign signifies that this force acts in a direction opposite to the displacement.
Now, velocity is ${{u}_{0}}$and the $\Delta y$ is the height of the fluid. So, the above relation becomes, $F=\dfrac{\eta A{{u}_{0}}}{h}$
It can be seen that force is directly proportional to the coefficient of viscosity, velocity, area and inversely proportional to the height.
Since the plate is moving with constant velocity, the same force must be acting on the floor.

So, Option (A), (C) and (D) are correct.

Note:
In this problem Since plate is moving with constant velocity, the same force must be acting on the floor. The viscous force is the force between a body and a fluid that is liquid or gas. Motion is must for viscous force to come into play.