Chapter 7, Problem 7.16P

Interpretation Introduction

Interpretation: The equation for the frictional moment in terms of other given factors is to be determined.

Concept introduction:

Frictional moment is the torque present due to the force of friction between any two moving objects.

The equation for the frictional moment is given as,

  $\text{M}\text{=}{\displaystyle \int \text{rdF}}$ ……. (1)

M = Frictional moment

r = Radius of the conical pivot

dF = Force due to friction

The frictional force is the force which inhibits the motion of the fluid/particle through the substance on which it is moving.

The frictional force is given as,

  $\text{dF}\text{=}\text{τ}\text{×}\text{dA}$ ……. (2)

F = Frictional force

dA = Differential Surface area of the cone

  $\text{τ}$ = Shear stress developed

The differential surface area of the cone is given as,

  $\text{dA}\text{=}\text{2πrdL}$ ……. (3)

r = Radius of the cone

dL = Differential slant height of the cone

The shear stress is given by the Newton’s law of viscosity in which the shear stress is directly proportional to the shear rate and the proportionality constant is viscosity.

The equation governing the above statement is,

  $\text{τ}\text{=}\text{μ}\frac{\text{du}}{\text{dy}}$ ……. (4)

  $\text{μ}$ = Viscosity of the fluid

  $\frac{\text{du}}{\text{dy}}$ = Shear rate