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Chapter 11: Problem 34

A solid gold bar is pulled up from the hold of the sunken RMS Titanic. (a) What happens to its volume as it goes from the pressure at the ship to the lower pressure at the ocean's surface? (b) The pressure difference is proportional to the depth. How many times greater would the volume change have been had the ship been twice as deep? (c) The bulk modulus of lead is one- fourth that of gold. Find the ratio of the volume change of a solid lead bar to that of a gold bar of equal volume for the same pressure change.

Short Answer

Expert verified
The volume of the gold bar increases as it is raised to the surface. If the ship was at twice its current depth, the volume change would have been twice. The volume change of a lead bar for the same pressure change is four times greater than that of a gold bar.

Step by step solution

01

Determine the volume change as the bar is raised

According to the general gas law, as pressure decreases, a material tends to expand. Therefore, the volume of the solid gold bar will increase as it goes from the higher pressure at the ship's depth to the lower pressure at the ocean's surface.
02

Calculate the volume change if the ship was twice as deep

The pressure difference is proportional to depth. So if the ship was twice as deep, the pressure difference and hence the volume change would have been twice as much.
03

Comparison of volume change of a lead bar to that of a gold bar

The bulk modulus of a substance is inversely proportional to the volume change for a given pressure change. Given that the bulk modulus of lead is one-fourth that of gold, the volume change for the same pressure change is four times greater in the lead than in the gold. Hence, the ratio of the volume change of a solid lead bar to that of a gold bar of equal volume for the same pressure change is 4:1.

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Key Concepts

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

Volume Change
When a solid material, like a gold bar, is subjected to variations in pressure, its volume will generally adjust accordingly. As the bar is raised from the depths of the ocean to the surface, it experiences a decrease in pressure. This decrease prompts the bar to expand. This expansion is due to the material's inherent property of trying to occupy more space as the external pressure lessens.

In relation to depth, the pressure a material encounters is proportional to how deep it is. So, if you imagine the Titanic resting at a greater depth, the pressure acting on the gold bar would naturally be higher. This would lead to a greater volume change upon being raised, as a higher pressure differential allows the material to expand more when brought to the surface. Thus, deeper means more dramatic expansion for the same material when raised.

This concept is not only essential for understanding bulk modulus but also plays a crucial role in areas such as designing underwater vessels or handling materials at different environmental pressures.
Pressure Difference
Pressure plays a significant role in material deformation and volume change. The pressure difference is particularly vital since it is directly proportional to the depth from which the gold bar is raised.

In essence, at greater depths, the external pressure is higher. This pressure is primarily due to the weight of the water column above the gold bar. Pressure is calculated as the force per unit area and increases as more water column acts on the object.

Imagine doubling the depth from which the gold bar is raised. The resulting pressure difference between the deep ocean's floor and the surface would also be doubled. Hence, the volume change experienced by the gold bar would be doubled as well.

This notion is vital when considering underwater expeditions or constructions requiring precise calculations of material expansion due to pressure alterations.
Material Properties
The properties of materials, especially in terms of how they respond to pressure changes, are dictated by characteristics like the bulk modulus. The bulk modulus indicates a material's resistance to uniform compression.

In the case of the gold and lead bars, their different responses to the same pressure change highlight the role of bulk modulus. Gold, with a higher bulk modulus than lead, will undergo less volume change compared to lead when subjected to the same pressure difference.

Specifically, the bulk modulus of lead is one-fourth that of gold. This means lead is more susceptible to volume change under external pressure changes. If you expose both gold and lead bars of equal size to the same pressure difference, the volume change in lead would be four times that of the gold.

Understanding these properties helps in material selection and engineering applications where materials undergo pressure variations, ensuring safety and efficiency in designs.

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Most popular questions from this chapter

A steel wire with radius \(r_{\text {steel }}\) has a fractional increase in length of \(\left(\Delta l / l_{0}\right)_{\text {steel }}\) when the tension in the wire is increased from zero to \(T_{\text {steel }}\). An aluminum wire has radius \(r_{\text {al }}\) that is twice the radius of the steel wire: \(r_{\mathrm{al}}=2 r_{\text {steel }} .\) In terms of \(T_{\text {steel }},\) what tension in the aluminum wire produces the same fractional change in length as in the steel wire?

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