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

5% of the body's volume is above water; about 40.63% above quicksand. You can breathe.

Step 1: Understanding Buoyancy in Water

To determine the fraction of the body's volume that floats above water, we use Archimedes' principle: the buoyant force equals the weight of the displaced fluid. Given the body's density is 0.95 times that of water, it implies that the body's density is \( \rho_b = 0.95 \rho_w \), where \( \rho_w \) is the density of water. When the body floats, the volume submerged \( V_s \) displaces a weight equal to its own. Thus, \( V_s \rho_w = V_t \rho_b \), where \( V_t \) is the total volume. Solving for the submerged volume fraction, \( \frac{V_s}{V_t} = \frac{\rho_b}{\rho_w} = 0.95 \).

Step 2: Calculating the Above-Water Volume Fraction

The fraction of the body's volume above water is simply the remainder of the total volume not submerged. Thus, \( \frac{V_a}{V_t} = 1 - \frac{V_s}{V_t} = 1 - 0.95 = 0.05 \). This means 5% of the body's volume is above the water surface.

Step 3: Understanding Buoyancy in Quicksand

Now consider floating in quicksand with a density 1.6 times that of water, \( \rho_q = 1.6 \rho_w \). Using the same principle, the volume submerged in quicksand, \( V_{qs} \rho_q = V_t \rho_b \). Substituting the densities, \( V_{qs} (1.6\rho_w) = V_t (0.95\rho_w) \), simplifying gives \( \frac{V_{qs}}{V_t} = \frac{0.95}{1.6} \approx 0.59375 \).

Step 4: Calculating the Above-Quicksand Volume Fraction

The fraction of the body's volume above the quicksand is \( \frac{V_{qa}}{V_t} = 1 - \frac{V_{qs}}{V_t} = 1 - 0.59375 \approx 0.40625 \). Therefore, approximately 40.63% of the body's volume is above the quicksand surface.

Step 5: Determining Breathing Ability

To assess breathing capability, consider that a significant portion of your body is above the quicksand (approx 40.63%). Thus, your head, and therefore airways, will likely be above the quicksand allowing for breathing.

Key Concepts

Archimedes' Principle

Archimedes' principle is a fundamental concept in fluid mechanics, which relates to buoyancy. It states that a body submerged in a fluid is buoyed up by a force equal to the weight of the fluid displaced by the body.
This principle helps explain why objects float or sink when placed in water or any other fluid.

• When an object is placed in a fluid, such as water, it experiences an upward force called the buoyant force.
• The strength of this buoyant force is equal to the weight of the fluid that the object displaces.

This principle is crucial for solving problems related to floating or sinking. For instance, when you float in a swimming pool, your body displaces a volume of water equal to its own weight, which stops you from sinking entirely.

Density

Density is a measure of how much mass is contained within a given volume. It is expressed as mass per unit volume, usually in units of kilograms per cubic meter (kg/m³) or grams per cubic centimeter (g/cm³). In simpler terms, it's how "heavy" or "light" an object is for its size.
Water, a common reference point, has a density of about 1,000 kg/m³.

• A substance with density less than water will likely float, like oil or ice.
• Conversely, a substance with a density greater than water will sink.

The exercise highlights that the human body's density is 0.95 times that of water, meaning it is slightly less dense than water. This means you will float just below the water's surface, with a small part of your body above the water.

Fluid Mechanics

Fluid mechanics is a branch of physics concerned with the behavior of fluids (liquids and gases) and the forces upon them. This field encompasses the study of both liquids and gases in motion and at rest.
Key aspects of fluid mechanics include:

• Understanding how fluids flow and behave when subjected to forces.
• The effects of pressure, temperature, and viscosity on fluid motion.

The solution discusses fluid mechanics principles to determine how much of a body floats above a liquid surface, influenced by the liquid's density. This is essential to comprehend the interaction between a body and the fluid it is submerged in, such as water or quicksand.

Quicksand

Quicksand is a fascinating natural phenomenon often associated with danger in movies and stories, but understanding its actual nature helps demystify it. Quicksand forms when water saturates an area of sand, leading to a mixture that behaves as a liquid, but it has a higher density than water.
Here's how it works:

• Water seeps into sand, forcing sand particles apart and reducing friction between them.
• This creates a "trap," which can support weight, but isn't as solid as it appears.

Interestingly, because quicksand is more dense than the human body (1.6 times more dense than water), you will float more in quicksand than in water. In practical terms, this means a larger portion of your body remains above the surface, allowing you to breathe while you are caught in it.