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Chapter 14: Problem 61

Advertisements for a certain small car claim that it floats in water. (a) If the car's mass is 900 \(\mathrm{kg}\) and its interior volume is 3.0 \(\mathrm{m}^{3}\) , what fraction of the car is immersed when it floats? You can ignore the volume of steel and other materials. (b) Water gradually leaks in and displaces the air in the car. What fraction of the interior volume is filled with water when the car sinks?

Short Answer

Expert verified
(a) 30% of the car is immersed when it floats. (b) 70% of the interior volume is filled with water when the car sinks.

Step by step solution

01

Determine the Volume Displaced by the Car

When the car floats, it displaces a volume of water equal to its weight. The weight of the car is given by \( W = m \cdot g = 900 \text{ kg} \times 9.8 \text{ m/s}^2 = 8820 \text{ N} \). The weight of the displaced water is equal to the weight of the car, so \( W = \rho \cdot V_{displaced} \cdot g \), where \( \rho = 1000 \text{ kg/m}^3 \) is the density of water. Solving for \( V_{displaced} \): \[ V_{displaced} = \frac{W}{\rho \cdot g} = \frac{8820 \text{ N}}{1000 \text{ kg/m}^3 \times 9.8 \text{ m/s}^2} = 0.9 \text{ m}^3 \]
02

Calculate the Fraction of the Car Immersed

The fraction of the car immersed in water when it is floating can be calculated by the formula:\[ \text{Fraction Immersed} = \frac{V_{displaced}}{V_{car}} = \frac{0.9 \text{ m}^3}{3.0 \text{ m}^3} = 0.3 \]
03

Determine Volume Displaced when Sinking

When the car sinks, it displaces a volume of water equal to its total volume, which is 3.0 \( \text{m}^3 \). This occurs when the weight of the entire displaced volume matches the car's overall weight plus the weight of any water inside.
04

Calculate the Required Water Volume Inside when Sinking

For the car to sink, the sum of its own weight plus the weight of water inside it must equal the buoyant force of 3.0 \( \text{m}^3 \) of displaced water. Let's calculate the buoyant force when completely submerged:\[ F_{buoyant} = \rho \cdot V_{car} \cdot g = 1000 \text{ kg/m}^3 \times 3.0 \text{ m}^3 \times 9.8 \text{ m/s}^2 = 29400 \text{ N} \]Set the total weight \( 9000 \text{ kg} + \text{water weight} \) equal to this buoyant force: Let the volume of water inside be \( V_{water} \), so the weight of this water is \( 1000 \cdot V_{water} \cdot 9.8 = 29400 - 8820 \). Solving for \( V_{water} \) gives\[ V_{water} = \frac{20580 \text{ N}}{1000 \cdot 9.8} = 2.1 \text{ m}^3 \]
05

Calculate the Fraction of the Interior Volume Filled with Water

The fraction of the interior volume that is filled with water when the car sinks is given by:\[ \text{Fraction Filled} = \frac{V_{water}}{V_{interior}} = \frac{2.1 \text{ m}^3}{3.0 \text{ m}^3} = 0.7 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Density of Water
The density of water is a key factor in understanding how objects behave when placed in water. Specifically, it is typically measured as 1000 kilograms per cubic meter (kg/m鲁). This means that each cubic meter of water has a mass of 1000 kilograms. Understanding the density of water is essential because it plays a crucial role in determining whether an object will float or sink. When submerged in water, an object experiences an upward force known as the buoyant force. This force is directly related to the density of the water that the object displaces. Water's density remains consistent under standard conditions, allowing for predictable calculations of buoyancy involving objects like the small car in our problem.
Buoyant Force
The buoyant force is the upward force a fluid exerts on an object placed in it. This force allows objects, such as boats or, in this case, a small car, to float or remain submerged at a certain level. The amount of buoyant force an object experiences depends on the fluid's density and the volume of fluid displaced by the object. When an object floats, the buoyant force equals the object's weight, maintaining it in equilibrium. For our floating car example, it displaces enough water to counterbalance its weight, which is 8820 Newtons. If an object sinks, the buoyant force is unable to balance the object's weight which means the total buoyant force, in this case of 29,400 Newtons when submerged, must match both the car's weight and the added weight of water filling its interior.
Archimedes' Principle
Archimedes' Principle is a fundamental law of physics that describes how and why objects float. It states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. This principle is crucial in analyzing floating and sinking objects. In applying Archimedes' Principle to our problem, when the car floats, it displaces a volume of water equal to 0.9 cubic meters, which equals its weight. This demonstrates the principle, as the buoyant force holding the car is derived from this displaced water weight. When the car begins to sink, Archimedes' Principle is again employed. As water fills the car, the principle guides us to understand why the vehicle eventually becomes fully submerged, requiring the displaced water's weight to counter both the car and the additional water weight, achieving equilibrium with the total buoyant force for sinking.

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Most popular questions from this chapter

Miles per Kilogram. The density of gasoline is 737 \(\mathrm{kg} / \mathrm{m}^{3}\) . If your new hybrid car gets 45.0 miles per gallon of gasoline, what is its mileage in miles per kilogram of gasoline? (See Appendix E.)

In a lecture demonstration, a professor pulls apart two hemispherical steel shells (diameter \(D\) ) with ease using their attached handles. She then places them together, pumps out the air to an absolute pressure of \(p\) , and hands them to a bodybuilder in the back row to pull apart. (a) If atmospheric pressure is \(p_{0}\) , how much force must the bodybuilder exert on each shell? (b) Evaluate your answer for the case \(p=0.025 \mathrm{atm}, D=10.0 \mathrm{cm} .\)

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