91Ó°ÊÓ

A useful measure of an individual's physical condition is the fraction of his or her body that consists of fat. This problem describes a simple technique for estimating this fraction by weighing the individual twice, once in air and once submerged in water. (a) A man has body mass \(m_{b}=122.5 \mathrm{kg} .\) If he stands on a scale calibrated to read in newtons, what would the reading be? If he then stands on a scale while he is totally submerged in water at \(30^{\circ} \mathrm{C}\) (specific gravity \(=0.996\) ) and the scale reads \(44.0 \mathrm{N},\) what is the volume of his body (liters)? (Hint: Recall from Archimedes' principle that the weight of a submerged object equals the weight in air minus the buoyant force on the object, which in turn equals the weight of water displaced by the object. Neglect the buoyant force of air.) What is his body density, \(\rho_{\mathrm{b}}(\mathrm{kg} / \mathrm{L}) ?\) (b) Suppose the body is divided into fat and nonfat components, and that \(x_{f}\) (kilograms of fat/kilogram of total body mass) is the fraction of the total body mass that is fat: \(x_{\mathrm{f}}=\frac{m_{\mathrm{f}}}{m_{\mathrm{b}}}\) Prove that \(x_{\mathrm{f}}=\frac{\frac{1}{\rho_{\mathrm{b}}}-\frac{1}{\rho_{\mathrm{nf}}}}{\frac{1}{\rho_{\mathrm{f}}}-\frac{1}{\rho_{\mathrm{nf}}}}\) where \(\rho_{\mathrm{b}}, \rho_{\mathrm{f}},\) and \(\rho_{\mathrm{nf}}\) are the average densities of the whole body, the fat component, and the nonfat component, respectively. [Suggestion: Start by labeling the masses ( \(m_{\mathrm{f}}\) and \(m_{\mathrm{b}}\) ) and volumes \(\left(V_{\mathrm{f}} \text { and } V_{\mathrm{b}}\right)\) of the fat component of the body and the whole body, and then write expressions for the three densities in terms of these quantities. Then eliminate volumes algebraically and obtain an expression for \(\left.m_{f} / m_{b} \text { in terms of the densities. }\right]\) (c) If the average specific gravity of body fat is 0.9 and that of nonfat tissue is \(1.1,\) what fraction of the man's body in Part (a) consists of fat? (d) The body volume calculated in Part (a) includes volumes occupied by gas in the digestive tract, sinuses, and lungs. The sum of the first two volumes is roughly \(100 \mathrm{mL}\) and the volume of the lungs is roughly 1.2 liters. The mass of the gas is negligible. Use this information to improve your estimate of \(x_{\mathrm{f}}\).

Step by step solution

01

Part (a) - Calculation of volume and density

From Newton’s second law, \(F = mg\), the force exerted by the man on the scale in air (Weight) is \(m_{b}g\). In water, the reading is the difference between the weight in air and the buoyant force, according to Archimedes' Principle. Let's denote the volume of the body as \(V_{b}\). The buoyant force equals to the weight of the water displaced. So, the weight of water displaced is \(m_{b}g - 44 N =1000 kg/m^3 * V_{b} * g\). Solve this equation to find \(V_{b}\). The body density \(\rho_{b}\) is then \(m_{b}/V_{b}\).

02

Part (b) - Prove algebraic expression

To prove the given expression, start by expressing quantities in terms of masses and volumes. Densities can be defined as \(\rho_{f} = m_{f}/V_{f}\), \(\rho_{nf} = m_{nf}/V_{nf}\), and \(\rho_{b} = m_{b}/V_{b}\). Here, the fat and non-fat masses add up to the total mass (\(m_{f} + m_{nf} = m_{b}\)) and their volumes add up to the total volume (\(V_{f} + V_{nf} = V_{b}\)). Now, substitute everything in terms of \(m_{f}\), \(m_{nf}\), and \(m_{b}\). Use the given equation for \(x_{f}\) and eliminate volumes algebraically. With some simplification, it will lead to the required equation to be proved.

03

Part (c) - Calculation of the fraction of body fat

We simply substitute the given values of \(\rho_{f} = 0.9 kg/L\), \(\rho_{nf} = 1.1 kg/L\), and \(\rho_{b}\) (calculated in part (a)) into the equation \(x_{f}\) proved in part (b). This will give the fraction of the body that consists of fat.

04

Part (d) - Adjusted calculation of the fraction of body fat

Given the additional volume of gas in the body, subtract this volume from the original body volume (\(V_{b}\)) in part (a). Recalculate body density \(\rho_{b}\) using this modified volume. Redo the calculation in part (c) using this revised density for \(\rho_{b}\). This will give an improved estimate of the fraction of fat \(x_{f}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Archimedes' Principle

Archimedes' principle is a fundamental concept in fluid mechanics, which states that any object, fully or partially submerged in a fluid, experiences an upward buoyant force that is equal to the weight of the fluid displaced by the object. This principle can be used to determine the volume of an object by submerging it in a fluid and measuring the buoyant force.

For instance, when a body is submerged in water, it loses apparent weight. This apparent weight loss is due to the buoyant force, which opposes the force of gravity. In our exercise, when the man weighing 122.5 kg is submerged in water, the scale reads a much lower value because the water's buoyant force is acting on him. The difference between his weight in air and underwater directly allows us to calculate his body's volume. This is crucial for measuring body density, which in turn is used for estimating body fat content, a key indicator of physical health.

Calculating Body Density

Body density calculation is an essential step in understanding body composition, particularly when estimating the fat content of a body. Calculating the body density involves measuring the mass of an individual and the volume their body occupies.

This calculation is often performed with the help of Archimedes' principle, by comparing the weight of the individual in the air versus the weight when submerged in water. The difference in these weights, considering the density of the water and the impact of gravity, leads us to the volume of the body. With the volume and mass (obtained from weighing the individual on a scale in air), we can calculate the body density as \( \rho_{\mathrm{b}} = \frac{m_{\mathrm{b}}}{V_{\mathrm{b}}} \).

Moreover, since different tissues in the body have varying densities, such as fat versus muscle or bone, comparing the overall body density to standard values can help determine the relative proportion of fat to non-fat tissue.

Estimating Fat Content

Estimating fat content in a body is crucial for assessing a person's health and fitness levels. Fat content estimation is based on comparing the overall body density with the densities of pure fat and non-fat components of the body.

In the context of our exercise, once a person's body density is known, we can estimate the fat content using a mathematical formula derived from principles of volume and mass conservation. The proposed formula for fat content estimation takes into account the average densities of the whole body, the fat component (\(\rho_{\mathrm{f}}\)), and the non-fat component (\(\rho_{\mathrm{nf}}\)). It calculates the fraction of the total body mass that is fat (\(x_{\mathrm{f}}\)).

This method of estimation is particularly valuable because it doesn't rely on external appearances or measurements, which can be misleading. Keeping in mind the presence of gas in the digestive tract and lungs, as the exercise suggests, adjustments to the measured body volume should be made for a more accurate estimation of the fat content.