Chapter 2, Problem 2.20P

Interpretation Introduction

Interpretation: The value of density for the given condition of log should be determined.

Concept introduction:

Density is defined as the ratio of mass per unit volume. For flow conditions, density varies largely for the case of gases.

  $\underset{\text{ΔV}\to \delta V}{\rho }=\frac{\Delta m}{\Delta V}$

Where,

  $\Delta m$ is the mass of the fluid.

  $\Delta V$ is the volume.

  $\delta V$ is the smallest change in volume

Buoyancy: It is defined as the upward forces exerted by fluid that opposes the weight of immersed object. It is based on Archimedes principle which states the value of upward force is equal to the weight of fluid displaced.

The formula for resultant force is given as,

  $F=(\rho -{\rho }_B)g{V}_{ey}$

Where,

F is the resultant force.

  $\rho g{V}_{ey}$ is the weight.

  ${\rho }_{B}g{V}_{ey}$ is the buoyant force.

The formula for buoyant force is given by,

  $F_{upward}=\frac{{\rho }_{w}gA}{2}$

Based on this principle, the value of downward force is the sum of weight of circular log with weight of water supported by log as barrier, thus the formula for downward force is,

F _downward= weight of circular log + weight of water supported by log as barrier

Weight of circular log = $\rho gA$

Weight of circular log = ${\rho }_{w}g(r^2-\frac{\pi r^2}{4})$

  $F_{downward}=\rho g\pi r^2+{\rho }_{w}g(r^2-\frac{\pi r^2}{4})$

Where,

  ${\rho }_w$ is the density of water.

  $g$ is the acceleration due to gravity.

A is the area of sector.