Problem 29 - University Physics with Modern Physics

Question

An object of average density \(\rho\) floats at the surface of a fluid of density \(\rho_{\text { fluid. }}\) (a) How must the two densities be related? (b) In view of the answer to part (a), how can steel ships float in water? (c) In terms of \(\rho\) and \(\rho\) fluid, what fraction of the object is submerged and what fraction is above the fluid? Check that your answers give the correct limiting behavior as \(\rho \rightarrow \rho_{\text { fluid }}\) and as \(\rho \rightarrow 0 .\) (d) While on board your your your cousin Throckmorton cuts a rectangular piece (dimensions \(5.0 \times 4.0 \times 3.0 \mathrm{cm}\) out of a life preserver and throws it into the ocean. The piece has a mass of 42 g. As it floats in the ocean, what percentage of its volume is above the surface?

Step-by-Step Solution

Answer

(a) \( \rho < \rho_{\text{fluid}} \); (b) Large air spaces reduce average density; (c) Submerge fraction is \( \frac{\rho}{\rho_{\text{fluid}}} \); (d) 32.1% volume above water.

Step 1: Relate Densities for Floating

For an object to float, its average density \( \rho \) must be less than the density of the fluid \( \rho_{\text{fluid}} \). This is because the force of buoyancy that counters the weight of the object depends on the fluid density. Mathematically, for an object to float without sinking, \( \rho < \rho_{\text{fluid}} \).

Step 2: Explanation for Steel Ships Floating

Steel is much denser than water, with density \( \rho_{\text{steel}} > \rho_{\text{water}} \). Steel ships float because their overall density, considering the air inside, is less than the density of water. The average density of the ship becomes less than the water when constructed properly to include large air-filled spaces.

Step 3: Submerged Fraction of the Object

If an object is floating at equilibrium, the weight of the fluid displaced equals the weight of the object. The fraction of the object submerged is given by the ratio \( \frac{\rho}{\rho_{\text{fluid}}} \), meaning the unsubmerged part is \( 1 - \frac{\rho}{\rho_{\text{fluid}}} \). As \( \rho \rightarrow \rho_{\text{fluid}} \), the object becomes fully submerged, and as \( \rho \rightarrow 0 \), it fully floats above the fluid.

Step 4: Calculate the Volume Percentage Above Surface

First, calculate the density of the piece of life preserver. Its mass is 42 g, and its volume is \( 5.0 \times 4.0 \times 3.0 = 60.0 \text{ cm}^3 \). Therefore, its density \( \rho = \frac{42}{60} = 0.7 \text{ g/cm}^3 \). Since the density of seawater (\( \rho_{\text{fluid}} \) ) is approximately \( 1.03 \text{ g/cm}^3 \), the fraction submerged is \( \frac{0.7}{1.03} \approx 0.679 \). Thus, the percentage of volume above water is \( 1 - 0.679 = 0.321 \), which is about 32.1%.

Key Concepts

Understanding Density in Buoyancy

Density is a key concept in understanding buoyancy, which is the ability of an object to float in a fluid. It is defined as the mass of an object divided by its volume, often expressed in units such as grams per cubic centimeter (g/cm³) or kilograms per cubic meter (kg/m³).
To determine if an object can float, we compare its density to that of the fluid in which it is placed:

• If the object's density \( \rho \) is less than the fluid's density \( \rho_{\text{fluid}} \), the object will float.
• If the object's density is equal to or greater than the fluid's density, the object will sink.

Think about a cork bobbing on water or a pebble sinking - it's all about the densities involved. Lower density materials, like the life preserver in the exercise, displace enough fluid equal to their weight before submerging fully, thus floating.

Exploring Fluid Mechanics

Fluid mechanics is the branch of physics that studies the behavior of fluids (liquids and gases) and the forces on them. One of its core principles involves understanding how fluids exert pressure and how external bodies interact with these forces. For objects in fluids:

• Pressure is exerted by the fluid on the object.
• The deeper an object is submerged, the greater the pressure.

A practical application is seen in ships, which are engineered to ensure that even materials denser than water, like steel, can float. This is achieved by designing the hull with a large volume, keeping the ship's overall density lower than that of the water. Factors such as balance, buoyancy distribution, and water displacement all play a role.
Fluid mechanics is behind the thought that, upon adding cabins filled with air, the ship effectively displaces a volume of water heavier than the ship itself, allowing it to float despite its dense materials.

Applying Archimedes' Principle

Archimedes' principle is fundamental in explaining buoyancy. It states that an object submerged in a fluid experiences an upward force equal to the weight of the fluid it displaces. This principle helps elucidate why the piece of life preserver in the exercise floats with part of it above the water.
When considering an object:

• If the weight of the displaced fluid is more than the object's weight, the object floats, partially submerged.
• If equal, the object remains submerged without sinking further.

For example, the life preserver's calculation shows that it displaces seawater with a weight equivalent to 67.9% of its own weight, causing the remaining 32.1% of its volume to remain above water. This aligns with Archimedes' principle, which perfectly explains the delicate balance achieved between density, fluid displacement, and buoyancy forces.