Problem 76 - Fundamentals of Physics

Question

Suppose that your body has a uniform density of 0.95 times that of water. (a) If you float in a swimming pool, what fraction of your body's volume is above the water surface? Quicksand is a fluid produced when water is forced up into sand, moving the sand grains away from one another so they are no longer locked together by friction. Pools of quicksand can form when water drains underground from hills into valleys where there are sand pockets. (b) If you float in a deep pool of quicksand that has a density 1.6 times that of water, what fraction of your body's volume is above the quicksand surface? (c) Are you unable to breathe?

Step-by-Step Solution

Answer

(a) 5% above water, (b) ~40.6% above quicksand, (c) Yes, you can breathe.

Step 1: Understanding Buoyancy

To solve this problem, let's apply Archimedes' principle, which states that the buoyant force on a submerged object is equal to the weight of the fluid displaced by the object. This means the weight of the water displaced by the body must equal the weight of the body itself.

Step 2: Calculating Fraction Above Water for (a)

Since the body's density is 0.95 times that of water, a fraction of the body 0.95 will be submerged in water when floating. The fraction of the volume above the water is then: \( \text{Fraction Above} = 1 - 0.95 = 0.05 \).

Step 3: Applying Archimedes’ Principle to Quicksand for (b)

For part (b), we have the density of quicksand as 1.6 times that of water. The ratio between the densities of the body and the quicksand is, therefore, \( \frac{0.95}{1.6} \). This ratio indicates the fraction of the body's volume that will be submerged, which is \( \frac{0.95}{1.6} ≈ 0.59375 \).

Step 4: Calculating Fraction Above Quicksand

From Step 3, the fraction of the body submerged is 0.59375. Thus, the fraction of your body's volume floating above the quicksand surface is: \( \text{Fraction Above} = 1 - 0.59375 ≈ 0.40625 \).

Step 5: Assessing Breathing Ability for (c)

Since more than 40% of your body's volume is above the quicksand surface, which is significantly more than needed to keep the head clear for breathing, you can indeed breathe while floating.

Key Concepts

Archimedes' Principle

Archimedes' Principle is a fundamental concept in fluid mechanics. It explains why objects float or sink in a fluid. It states that an object submerged in a fluid experiences a buoyant force that is equal to the weight of the fluid it displaces. This principle is vital for understanding buoyancy in both liquids like water and non-Newtonian fluids like quicksand.

Consider a boat on water: it floats due to the buoyant force equal to the weight of the water displaced by its hull. Similarly, if you are floating in a swimming pool, your body displaces a certain volume of water, and the buoyant force arising from this displacement helps keep you afloat. Remember, if the weight of the fluid displaced is greater than the weight of the object, it will float.

For our exercise, Archimedes' Principle allows us to determine how much of an object remains underwater or above water by comparing the density of the object to the density of the fluid it is in. This concept becomes especially interesting when considering fluids with varying densities, such as quicksand.

Density

Density is a measure of how much mass is contained in a given volume. It is represented by the formula: \( \rho = \frac{m}{V} \), where \( \rho \) is the density, \( m \) is the mass, and \( V \) is the volume. Density plays a crucial role in determining whether an object will float or sink when placed in a fluid.

An object's ability to float depends on its density relative to the fluid's density. If the object's density is less than the fluid's density, it will float; if it is greater, the object will sink. For example, in our exercise, the density of the body is 0.95 times that of water. This lower density allows a small portion of the body to remain above the water's surface when floating.

In contrast, quicksand has a higher density, being 1.6 times that of water. A body that has a density lower than quicksand, like ours, will have a significant portion above the quicksand's surface due to the submerged volume only needing to displace an amount equal to the body's weight.

Floating and Submersion

Floating and submersion relate closely to buoyancy and density. When any object is placed in a fluid, a judgment call between floating and submersion depends on whether the buoyant force can counteract the object's weight.

For floating, part of the object remains above the fluid's surface. It means the buoyant force equals the object's weight but not all of the object needs to be under the fluid. Our exercise shows that with a density of 0.95 times that of water, about 5% of the body stays above water.

If submersion occurs, it means the buoyant force could not keep the object on the surface, often resulting in it being totally below the fluid level. However, with quicksand being denser than water, more of the body's volume will float above the quicksand, giving us a 40.625% above-surface reading, due to the buoyant force managing to surpass the body's weight considerably.

Fluid Mechanics

Fluid Mechanics is a vast field of physics that deals with the behavior of fluids and their interactions with various forces. Understanding buoyancy through Archimedes' Principle is just one part of this field. It also touches on how fluids like quicksand behave compared to typical fluids like water.

Quicksand is an interesting example of fluid mechanics. It behaves both as a liquid and a solid due to changes in pressure. When pressure is applied, it transforms from a seemingly solid surface into a fluid, significantly altering its effective density. Fluid mechanics helps explain how you can float in quicksand, with higher density creating a more significant buoyant force due to greater displacement per volume of the buoyant substance.

This knowledge underscores the importance of calculating density such as whether an object will float in various fluids. Additionally, it aids in understanding more complex systems, like the formation and behavior of quicksand, cementing its significance in both theoretical and practical applications.