Chapter 3, Problem 3.15P

Interpretation Introduction

(a)

Interpretation:

Calculate the body weight and the body volume of the man.

Concept introduction:

Archimedes’ Principle is defined as:

The upward buoyant force that is exerted on a body immersed in a fluid is equal to the weight of the fluid that the body displaces or weight of the body immersed in a fluid.

This can be rewritten as,

Buoyant force (up) = Weight of block (down)

The specific gravity of the liquid is 0.996.

Interpretation Introduction

(b)

Interpretation:

Assume the body can be divided into nonfat and fat components. Derive an expression for the ratio of the mass of fat to mass of the whole body.

Concept introduction:

Assume the body can be divided into nonfat and fat components and the following notations are defined,

Fraction of total body mass that is fat = $x_f$

Average density of the whole body = ${\rho }_b$

Density of the fat component = ${\rho }_f$

Density of the nonfat component = ${\rho }_{nf}$

Mass of the fat component = $m_f$

Mass of the body = $m_b$

Volume of the fat component = $V_f$

Volume of the body = $V_b$

This problem can be solved by, first derive expressions for the three densities then eliminate volumes algebraically so you can approach the final answer.

Interpretation Introduction

(c)

Interpretation:

Calculate the fraction of fat in a human body for the given case.

Concept introduction:

Assume the body can be divided into nonfat and fat components. The fraction of fat can be calculated using the following relationship.

$x_f=\frac{1/{\rho }_b-1/{\rho }_{nf}}{1/{\rho }_f-1/{\rho }_{nf}}$

Where,

Fraction of total body mass that is fat = $x_f$

Average density of the whole body = ${\rho }_b$

Density of the fat component = ${\rho }_f$

Density of the nonfat component = ${\rho }_{nf}$

Interpretation Introduction

(d)

Interpretation:

Improve the estimation of $x_f$ for the given case.

Concept introduction:

Assume the body can be divided into nonfat and fat components. The fraction of fat can be calculated using the following relationship.

$x_f=\frac{1/{\rho }_b-1/{\rho }_{nf}}{1/{\rho }_f-1/{\rho }_{nf}}$

Where,

Fraction of total body mass that is fat = $x_f$

Average density of the whole body = ${\rho }_b$

Density of the fat component = ${\rho }_f$

Density of the nonfat component = ${\rho }_{nf}$

The above relationship includes the gas in tract, sinuses and lungs. The total volume is 1.4 liters. Assume that the mass of gas is negligible.