Problem 30 - College Physics

Question

The average human has a density of \(945 \mathrm{~kg} / \mathrm{m}^{3}\) after inhaling and \(1020 \mathrm{~kg} / \mathrm{m}^{3}\) after exhaling. (a) Without making any swimming movements, what percentage of the human body would be above the surface in the Dead Sea (a body of water with a density of about \(1230 \mathrm{~kg} / \mathrm{m}^{3}\) ) in each of these cases? (b) Given that bone and muscle are denser than fat, what physical characteristics differentiate "sinkers" (those who tend to sink in water) from "floaters" (those who readily float)?

Step-by-Step Solution

Answer

After inhaling, around 76.83% of the human body would be submerged in the Dead Sea. After exhaling, this increases to around 82.93%. 'Sinkers' generally have more muscle and bone which are denser than fat, whereas 'floaters' have a greater proportion of fat.

Step 1: Calculate the percentage of the human body above the surface after inhaling

The percentage of a human body above the surface is determined by the ratio of the density of the human body after inhaling (\(945 \mathrm{kg/m^3}\)) to the density of the Dead Sea (\(1230 \mathrm{kg/m^3}\)). This ratio is given by \[\% \text{above surface} = \frac{\text{Density of human body}}{\text{Density of Dead Sea}}* 100\% = \frac{945 \mathrm{kg/m^3}}{1230 \mathrm{kg/m^3}}* 100\%.\]

Step 2: Calculate the percentage of the human body above the surface after exhaling

Now, we repeat the same process for the human body after exhaling, which has a higher density (\(1020 \mathrm{kg/m^3}\)), than after inhaling. We therefore get \[\% \text{above surface} = \frac{\text{Density of human body after exhaling}}{\text{Density of Dead Sea}}* 100\% = \frac{1020 \mathrm{kg/m^3}}{1230 \mathrm{kg/m^3}}* 100\%.\]

Step 3: Observing physical characteristics to differentiate 'sinkers' from 'floaters'

'Sinkers' are individuals who tend to be denser than water, that is, they have more muscle and bone which are denser than fat. 'Floaters', on the other hand, have a greater proportion of fat, making them less dense than water. Thus, the physical trait that separates 'sinkers' from 'floaters' is the relative proportions of fat, muscle and bone.

Key Concepts

Density

Density is a key concept in understanding buoyancy and why certain objects float while others sink. It is defined as mass per unit volume and is often expressed in kilograms per cubic meter (kg/m³). In simpler terms, density tells us how much stuff is packed into a space.

• For example, if you have a lot of mass in a small volume, that object has a high density, like a rock.
• If the mass is spread out in a large volume, it has a low density, like a sponge.

When it comes to floating or sinking, the density of an object compared to the liquid it is in plays a critical role.

• If the object's density is less than the liquid's, it will float.
• If it's more, it will sink.

For humans, density varies slightly when breathing. Inhaling fills the lungs with air, reducing overall body density. Exhaling reduces the air in the lungs, increasing body density.

Floating and Sinking

Floating and sinking are phenomena that occur due to the buoyant force, which acts against the weight of an object in a fluid. This buoyant force comes from differences in density.
To visualize this, you can think of objects being pressed by the fluid from all sides. If the upward force is greater than or equal to the object's weight, it floats; if not, it sinks.

• A person is more likely to float in water if their overall density is lower than the water’s.
• The action of lungs helps because by inhaling, more air is introduced, increasing buoyancy as the density decreases.

The concept can also be used to explain why the same person floats more easily in the Dead Sea than in usual seawater. It is all about the body's density relative to the density of the water they are in.

Dead Sea

The Dead Sea is famous for its high salinity, which makes its water denser than typical freshwater or seawater. This density is approximately 1230 kg/m³, which is much higher than most human body densities when breathing normally.

• Due to its high density, the buoyant force exerted by Dead Sea water is stronger, making it much easier for a person to float.
• This means even people who typically "sink" in normal water can float effortlessly in the Dead Sea.

The phenomenon is so distinct that your body will float even if you lie flat without moving your arms or legs. This unique feature draws many visitors to experience the sensation of floating without effort. Understanding the Dead Sea's dense waters gives an excellent illustration of the relationship between density and buoyancy.