Problem 73 - Fundamentals of Physics

Question

About one-third of the body of a person floating in the Dead Sea will be above the waterline. Assuming that the human body density is \(0.98 \mathrm{~g} / \mathrm{cm}^{3}\), find the density of the water in the Dead Sea. (Why is it so much greater than \(1.0 \mathrm{~g} / \mathrm{cm}^{3} ?\) )

Step-by-Step Solution

Answer

The density of Dead Sea water is approximately \(1.47 \ \mathrm{g/cm^3}\) due to its high salt content.

Step 1: Understand Buoyancy

According to the principle of buoyancy, for a floating object, the weight of the fluid displaced by the submerged part of the object is equal to the weight of the object. This means the average density of the human body times its volume must equal the water density times the submerged volume.

Step 2: Express Given Information Mathematically

Given that one-third of the body is above the waterline, two-thirds (\(\frac{2}{3}\)) is submerged. So, if the body's density is \(0.98 \ \mathrm{g/cm^3}\), we can set up the equation for the densities: \( \rho_{\text{body}} \times V = \rho_{\text{water}} \times \left(\frac{2}{3}V\right) \)

Step 3: Solve for Water Density

Rearrange the equation from Step 2 to find the water density: \( \rho_{\text{water}} = \frac{\rho_{\text{body}}}{\frac{2}{3}} = \frac{0.98}{\frac{2}{3}} \approx 1.47 \ \mathrm{g/cm^3} \).

Step 4: Conclusion on Density of Dead Sea Water

The calculated density of \(1.47 \ \mathrm{g/cm^3}\) indicates a higher density than freshwater, which explains the buoyancy effect of floating in the Dead Sea. The increased density is due to the high salt concentration in Dead Sea water.

Key Concepts

Density

Density is an essential concept in physics, representing how much mass is contained in a given volume. It is commonly measured in units like grams per cubic centimeter (g/cm³). Generally, density is a way to express how tightly matter is packed within a space. Math shows us this in the formula: \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). A high-density object has more mass packed into each unit of volume, whereas a low-density object has less mass in the same volume. Understanding density is crucial when studying buoyancy, as it determines whether an object will float or sink in a fluid.

Principle of Buoyancy

The principle of buoyancy, also known as Archimedes' principle, is a fascinating aspect of physics. It states that an object submerged in a fluid (like water) experiences an upward force called "buoyancy," which is equal to the weight of the fluid that the object displaces. This principle helps us understand why objects float or sink. When an object floats, the buoyant force is equal to the weight of the object. The part of the object submerged under the surface displaces a volume of fluid that weighs as much as the object itself. Hence, objects with densities lower than the fluid will float. If they are denser than the fluid, they will sink.

Density of Water

Water is often used as a reference point when discussing density. The typical density of fresh water is about 1.0 g/cm³ at room temperature. This constant value makes it easier to determine whether other substances will float or sink. The density of water can change slightly under different temperatures and conditions. Water expands upon heating and contracts when cooled, affecting the density. Yet, in many practical applications, we consider it to be 1.0 g/cm³. In the Dead Sea, however, the water density is significantly higher due to dissolved minerals and salts, altering the usual pattern of floating and sinking.

Floating

Floating is an interesting phenomenon where an object remains on the surface of a fluid. It occurs when the object's overall density is less than the fluid's density. For a human in the Dead Sea, a third of their body stays above water, illustrating how density affects buoyancy. Here, the human body density of 0.98 g/cm³ is less than the Dead Sea water's density, making it easier to float. Objects less dense than the fluid they are in will float effortlessly, supported by the upward buoyant force that equals the object's weight.

Salt Concentration

Salt concentration significantly impacts the density of water. When salt is added to water, the density of the water increases because the mass increases while the volume remains nearly the same. The Dead Sea is a perfect example of high salt concentration. Its waters are filled with various salts and minerals, making it much denser than ordinary sea water or freshwater. With a density around 1.47 g/cm³, it's easier to float on its surface. This high concentration of salt not only helps in buoyancy but also contributes to the unique ecosystem and health benefits associated with the Dead Sea.