Question

The average density of the human body is \(985 \mathrm{~kg} / \mathrm{m}\), and typical density of seawater is about \(1024 \mathrm{~kg} / \mathrm{m}^{3}\), a) Draw a free-body diagram of a human body floating in seawater and determine what percentage of the body's volume is submerged. b) The average density of the human body, after maximum inhalation of air, changes to \(9.45 \mathrm{~kg} / \mathrm{m}^{3}\). As a person floating in seawater inhales and exhales slowly, what percentage of his volume moves up out of and down into the water? c) The Dead Sea (a saltwater lake between Israel and Jordan) is the world's saltiest large body of water. Its average salt content is more than six times that of typical seawater, which explains why there is no plant and animal life in it. Two-thirds of the volume of the body of a person floating in the Dead Sea is observed to be submerged. Determine the density (in \(\mathrm{kg} / \mathrm{m}^{3}\) ) of the seawater in the Dead Sea.

Short answer

Answer: The density of the Dead Sea water is approximately 1477.5 kg/m³. Two-thirds (or about 66.67%) of a human body's volume is submerged in the Dead Sea water.

Step-by-step solution

a) Free-Body Diagram and Percentage of Body's Volume Submerged

To draw a free-body diagram, we need to consider the forces acting on the human body floating in seawater. There are two forces: 1. The gravitational force (weight) acting downwards: \(F_w = mg\) 2. The buoyant force (upward force due to displaced water), given by Archimedes' principle: \(F_b = ρ_wV_wg\), where \(V_w\) is the volume of the displaced water. For a floating body, the buoyant force balances the weight of the body: \(F_b = F_w\). The volume of the displaced water depends on the percentage of submerged human body volume. From the mentioned principle, we have \(mg = ρ_wV_wg\). Cancelling out g from both sides gives \(m = ρ_wV_w\). The percentage of the human body's volume submerged is given by the ratio of the volumes: \(%submerged = \dfrac{V_w}{V} * 100\) Where \(V\) is the human body's volume. Using the average density of the human body (\(ρ_b = \dfrac{m}{V}\)), we can solve the equation for the percentage: \(ρ_bV = ρ_wV_w\) \(V_w = \dfrac{ρ_b}{ρ_w}V\) \(%submerged = \dfrac{ρ_b}{ρ_w} * 100 = \dfrac{985}{1024} * 100 ≈ 96.2\%\)

b) Change in Percentage Volume After Inhaling

After inhaling, the average density of the human body changes to \(ρ_b' = 9.45~\mathrm{kg/m^3}\), while the body's mass remains the same. Using this new density, we can find the new volume of the body, \(V'\): \(V' = \dfrac{m}{ρ_b'}\) The displaced water volume after inhaling is: \(V_w' = \dfrac{ρ_b'}{ρ_w}V'\) The change in percentage of volume is given by the difference between the volumes after and before inhaling: \(Δ% = \dfrac{V_w'}{V'} * 100 - \dfrac{V_w}{V} * 100\) \(Δ% = \dfrac{ρ_b'}{ρ_w} * 100 - \dfrac{ρ_b}{ρ_w} * 100 = \dfrac{9.45}{1024} * 100 - \dfrac{985}{1024} * 100 ≈ -95.1\%\)

c) Density of Seawater in the Dead Sea

Since two-thirds of the human body's volume is submerged in the Dead Sea, we can use this information to find its density. \(%submerged_{DeadSea} = \dfrac{2}{3}\) Using Archimedes' principle, we can write the equation for the volume of displaced water, \(V_{ds}\), in the Dead Sea: \(m = ρ_{ds}V_{ds}g\) The volume ratio is given by: \(\dfrac{V_{ds}}{V} = \dfrac{2}{3}\) Which implies, \(V_{ds} = \dfrac{2}{3}V\) Now, we have: \(m = ρ_{ds}(\dfrac{2}{3}V)g\) Substituting the body's density (\(ρ_b = \dfrac{m}{V}\)), we get: \(ρ_bV = ρ_{ds}(\dfrac{2}{3}V)g\) Solving for the density of the Dead Sea seawater (\(ρ_{ds}\)): \(ρ_{ds} = \dfrac{3}{2}ρ_b = \dfrac{3}{2}(985) = 1477.5~\mathrm{kg/m^3}\)

Key concepts

Buoyant Force

When an object is immersed in a fluid, such as seawater, it experiences an upward force known as the buoyant force. This force is vital for understanding why objects float or sink. Archimedes' principle states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. The formula to calculate this force is expressed as F_b = ρ_wV_wg, where ρ_w is the density of the fluid, V_w is the volume of fluid displaced, and g is the acceleration due to gravity.

For objects floating, such as the human body in seawater, the buoyant force exactly balances the object's weight. If the object's density is less than that of the fluid, it will float partly submerged until the buoyant force equals the object's weight. Understanding this concept allows us to determine what percentage of an object will be submerged when floating in a fluid with a known density, just as we did when calculating the volume of the human body submerged in seawater.

Density of Seawater

Density is a physical property defined as mass per unit volume, represented by the equation ρ = m/V. Seawater density varies slightly due to differences in salinity and temperature, but average seawater density is about 1024 kg/m³. This is crucial information when applying Archimedes' principle because the force exerted by a fluid on an object depends on the fluid’s density.

In specialized scenarios, like the Dead Sea, the density can increase substantially due to higher salinity levels, as indicated by the fact that two-thirds of a human body can be submerged, reflecting a higher buoyant force. The Dead Sea's six times higher salt content compared to average seawater not only influences the buoyancy but also the ecosystem, as its harsh environment is not conducive to most plant and animal life. Knowing the density of seawater in different contexts allows for accurate predictions and applications of Archimedes' principle.

Free-Body Diagram

A free-body diagram is a visual representation used to analyze the forces acting upon an object. For an object floating in a fluid such as seawater, we typically draw two forces: the object's weight pulling downwards and the buoyant force pushing upwards.

The weight is given by the formula F_w = mg, and the buoyant force, following Archimedes' principle, is F_b = ρ_wV_wg. In a free-body diagram for a floating human, we depict these forces with arrows pointing in opposite directions, scaled appropriately to indicate their magnitudes. If the object is stable and floating, these forces will be of equal magnitude and opposite direction, demonstrating equilibrium. Through free-body diagrams, students can visually parse out the physical principles at play, aiding in the understanding of concepts like equilibrium and buoyant force.