Question

A plate of metal $100c{m^2}$ in area rests on a layer of castor oil $2mm$ thick whose coefficient of viscosity is $15.5$ poise. The horizontal force required to move the plate with a uniform speed of $3cm{s^{ - 1}}$ is
A. $28.250$ dyne
B. $28,250$ Newton.
C. $23,250$ dyne
D. $28,250$ Newton

Answer 1

Hint: Use the formula of force in terms of viscosity to solve the above question. Make sure you check the dimensions of every quantity before substituting their values in the formula. All the units should belong to the same system of units.

Formula Used: $F = \eta A\dfrac{{dv}}{{dy}}$
Where,
$F$ is force
$\eta $ is the coefficient of viscosity
$dv$ is the velocity
$dy$ is thickness of the coil

Complete step by step answer:We know that horizontal force is required to move the plate.
The force can be defined as an interaction that, when unopposed, will change the motion of an object.
Equation of force in terms of viscosity is given by
$F = \eta A\dfrac{{dv}}{{dy}}$
$F$ is force
$\eta $ is the coefficient of viscosity
$dv$ is the velocity
$dy$ is thickness of the coil
It is given that,
Coefficient viscosity is $\mu = 15.5$ poise
Area of the metal plate is $A = 100c{m^2}$
Thickness of the coil is $dy = 2mm = 0.2cm$
Now, the plate is at rest in the beginning and then a force is applied to move it with the velocity of $3cm{s^{ - 1}}$.
Therefore, the change in velocity will be given as,
$dv = v - u$
$ = 3 - 0$
$ \Rightarrow dv = 3cm{s^{ - 1}}$
Therefore, we get
$\dfrac{{dv}}{{dy}} = \dfrac{3}{{0.2}}$
$ \Rightarrow \dfrac{{dv}}{{dy}} = 15c{m^{ - 1}}$
Therefore, $F = \eta A\dfrac{{dv}}{{dy}}$ will give
$F = 15.5 \times 100 \times 15dyne$
$ \Rightarrow F = 23250dyne.$
Therefore, from the above explanation, the correct option is (C) $23,250$dyne.

Note:This is a simple question of substituting the values in the formula. The only thing you have to observe here is the units of all the quantities. We could observe that all the units were in c.g.s. systems instead of SI units. Therefore, we converted the thickness into cm instead of meter. And that is why the force is in dyne instead of Newton.