A solid gold bar is pulled up from the hold of the sunken RMS Titanic. 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.

Verified step by step guidance

1. Understand the concept of bulk modulus, which is a measure of a material's resistance to uniform compression. It is defined as the ratio of the pressure change to the relative volume change: \( B = -\frac{\Delta P}{\frac{\Delta V}{V}} \).
2. Recognize that the problem involves comparing the volume change of two materials, lead and gold, under the same pressure change. The bulk modulus of lead is given as one-fourth that of gold.
3. Express the bulk modulus of lead \( B_{\text{lead}} \) in terms of the bulk modulus of gold \( B_{\text{gold}} \): \( B_{\text{lead}} = \frac{1}{4} B_{\text{gold}} \).
4. Use the formula for bulk modulus to express the volume change for each material. For gold, \( \Delta V_{\text{gold}} = -\frac{\Delta P \cdot V}{B_{\text{gold}}} \) and for lead, \( \Delta V_{\text{lead}} = -\frac{\Delta P \cdot V}{B_{\text{lead}}} \).
5. Substitute \( B_{\text{lead}} = \frac{1}{4} B_{\text{gold}} \) into the expression for \( \Delta V_{\text{lead}} \) and find the ratio \( \frac{\Delta V_{\text{lead}}}{\Delta V_{\text{gold}}} \). Simplify the expression to find the ratio of the volume change of lead to that of gold.

Key Concepts

Bulk Modulus

Bulk modulus is a measure of a material's resistance to uniform compression. It is defined as the ratio of pressure increase to the relative decrease in volume. A higher bulk modulus indicates that a material is less compressible. In this problem, understanding the bulk modulus helps compare how lead and gold respond to pressure changes.

Volume Change Under Pressure

Volume change under pressure refers to how the volume of a material changes when subjected to external pressure. This concept is crucial for determining how different materials, like lead and gold, will compress under the same pressure. The relationship between pressure, bulk modulus, and volume change is key to solving the problem.

Ratio of Volume Change

The ratio of volume change compares the extent to which two materials change in volume under identical pressure conditions. Given that lead's bulk modulus is one-fourth that of gold, this concept helps calculate how much more lead will compress compared to gold, using the inverse relationship between bulk modulus and volume change.