Chapter 4, Problem 4.8P

Interpretation Introduction

Interpretation: Velocity and acceleration at one end of a piston is to be calculated when values of velocity and acceleration at another end is given.

Concept introduction: The continuity equation is applied to calculate velocity and acceleration whenever solid in which some fluid is flowing is given and diameter at 2 ends of solid is different.

The continuity equation can be given as,

  ${\text{A}}_{\text{1}}{\text{v}}_{\text{1}}{\text{= A}}_{\text{2}}{\text{v}}_{\text{2}}$ ....... (1)

Here, A₁ and A₂ are the areas at 2 points of the solid body and are given by the following formula,

  $\begin{array}{l}{\text{A}}_{\text{1}}\text{=}\frac{\text{π}}{\text{4}}{\text{D}}_{\text{1}}{}^{\text{2}}\mathrm{.......}\text{(2)}\\ {\text{D}}_{\text{1}}\text{=}\text{Diameter}\text{at}\text{point}\text{1}\\ {\text{A}}_{\text{2}}\text{=}\frac{\text{π}}{\text{4}}{\text{D}}_{\text{2}}{}^{\text{2}}\mathrm{.......}\text{(3)}\\ {\text{D}}_{\text{2}}\text{=}\text{Diameter}\text{at}\text{point}\text{2}\end{array}$

The velocity at 2 points of the piston are ${\text{v}}_{\text{1}}{\text{ and v}}_{\text{2}}$ .

The acceleration at 2 points are ${\text{a}}_{\text{1}}{\text{ and a}}_{\text{2}}$ and their values can be calculated by the continuity equation just by replacing velocity by an acceleration in equation (1).

  ${\text{A}}_{\text{1}}{\text{a}}_{\text{1}}{\text{= A}}_{\text{2}}{\text{a}}_{\text{2}}$ ....... (4)