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Chapter 12: Problem 63

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. (b) The car sinks when 100% of its interior volume is filled with water.

Step by step solution

01

Understand Archimedes' Principle

When an object floats, the weight of the displaced fluid equals the weight of the object due to Archimedes' principle. The weight of water displaced by the car is equal to the weight of the car itself.
02

Calculate the Weight of the Car

Calculate the weight of the car using the formula: \( W = m \times g \), where \( m = 900\, \mathrm{kg} \) and \( g = 9.8\, \mathrm{m/s^2} \). The weight \( W = 900 \times 9.8 = 8820\, \mathrm{N} \).
03

Calculate the Buoyant Force

The buoyant force is equal to the weight of the displaced water, which is given by the volume of immersed part of the car times the density of water and gravity: \( F_b = V_{immersed} \times \rho_{water} \times g \), where \( \rho_{water} = 1000\, \mathrm{kg/m^3} \).
04

Set the Weight of Car Equal to Buoyant Force

For floating, \( 8820 = V_{immersed} \times 1000 \times 9.8 \). Solve for \( V_{immersed} = \frac{8820}{9800} = 0.9\, \mathrm{m^3} \).
05

Calculate Fraction of Immersed Volume

The fraction of the car that is immersed is \( \frac{V_{immersed}}{V_{total}} = \frac{0.9}{3.0} = 0.3 \). Thus, 30% of the car is immersed.
06

Evaluate Sinking Conditions

For the car to sink, the density of the car must be equal to the density of water when the interior is fully or partially filled with water. The total mass \( 900\, \mathrm{kg} \) remains constant, but the new volume causing drowning will be equal to the displaced volume.
07

Calculate Fraction of Volume Filled When Sinking

The car sinks when \( 900 = 900 \times \frac{V_{filled}}{3.0} \), giving \( V_{filled} = 3.0\, \mathrm{m^3} \). This means \( \frac{V_{filled}}{3.0} = 1.0 \). Hence, when the interior volume fills completely, the car will sink.

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Key Concepts

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

Buoyant Force
Buoyant force is a fundamental concept in fluid mechanics. It鈥檚 the force exerted by a fluid that opposes an object's weight. Imagine you are holding a ball underwater. The upward push you feel is the buoyant force. This force allows objects that are less dense than the fluid to float. Archimedes' Principle tells us the buoyant force on a submerged object is equal to the weight of the fluid it displaces.
  • The formula is: \( F_b = V_{immersed} \times \rho_{water} \times g \), where \( \rho_{water} \) is the density of water.
  • Here, \( V_{immersed} \) is the volume of the object submerged in the fluid.
Applying this concept, we see that if the buoyant force equals the gravitational force on the car, it will float. By investigating the balance of these forces, we determine how much of the car is submerged while it floats.
Density of Water
The density of water is a crucial factor in determining buoyancy. Density is defined as mass per unit volume, denoted by \( \rho \). For water, this value is approximately \( 1000 \, \mathrm{kg/m^3} \) at standard temperature and pressure conditions. Water's density lets us calculate the buoyant force for any object in water.
  • A high-density object displaces water and sinks.
  • A low-density object results in less water being displaced and thus floats.
Understanding this, when a car claims to float, it will only float if its average density, including its interior volume of air, is less than or equal to the density of water. Hence, knowing the density of water helps us predict whether the car will float or not.
Immersed Volume
Immersed volume is the part of the object that is under the surface of the fluid. It depends on the object's density relative to the fluid. For the car problem, finding the immersed volume involves solving for how much of the car's volume is submerged when it floats. Using the formula for buoyant force,
  • \( F_b = V_{immersed} \times \rho_{water} \times g \)
  • We found \( V_{immersed} = 0.9 \, \mathrm{m^3} \)
This means 0.9 cubic meters of the car's volume is under water. With the car's total volume being 3 cubic meters, only 30% is immersed while floating. This helps us understand how objects interact with water surfaces and allows predictions about their floating or sinking behavior.
Sinking Conditions
Sinking conditions of an object, such as the car in this scenario, depend on its density compared to the fluid it is in. An object will sink if its density becomes greater than the fluid's density. As water enters the car, its density will eventually equal the water鈥檚 density, leading to its sinking.
  • Sinking begins when the internal water volume is significant enough to alter the car's density.
  • The car stays afloat with the air-filled interior, but when this is displaced by water, sinking occurs.
In this example, when water fills the car, making the car behave as if it were one with the water's density, full volume is used. Our exercise showed that once the interior is completely filled with water, the car will inevitably sink. Understanding this helps with grasping broader fluid dynamics and floating principles.

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Most popular questions from this chapter

A cubical block of wood 0.100 \(\mathrm{m}\) on a side and with a density of 550 \(\mathrm{kg} / \mathrm{m}^{3}\) floats in a jar of water. Oil with a density of 750 \(\mathrm{kg} / \mathrm{m}^{3}\) is poured on the water until the top of the oil layer is 0.035 \(\mathrm{m}\) below the top of the block. (a) How deep is the oil layer? (b) What is the gauge pressure at the block's lower face?

You are preparing some apparatus for a visit to a newly discovered planet Caasi having oceans of glycerine and a surface acceleration due to gravity of 4.15 \(\mathrm{m} / \mathrm{s}^{2} .\) If your appara- tus floats in the oceans on earth with 25.0\(\%\) of its volume submerged, what percentage will be submerged in the glycerine oceans of Caasi?

Water stands at a depth \(H\) in a large, open tank whose side walls are vertical (Fig. \(\mathrm{Pl} 2.89 ) .\) A hole is made in one of the walls at a depth \(h\) below the water surface. (a) At what distance \(R\) from the foot of the wall does the emerging stream strike the floor? (b) How far above the bottom of the tank could a second hole be cut so that the stream emerging from it could have the same range as for the first hole?

At one point in a pipeline the water's speed is 3.00 \(\mathrm{m} / \mathrm{s}\) and the gauge pressure is \(5.00 \times 10^{4}\) Pa. Find the gauge pressure at a second point in the line, 11.0 \(\mathrm{m}\) lower than the first, if the pipe diameter at the second point is twice that the first.

A small circular hole 6.00 mm in diameter is cut in the side of a large water tank, 14.0 m below the water level in the tank. The top of the tank is open to the air. Find (a) the speed of efflux of the water and (b) the volume discharged per second.
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