The Density and Volume Displacement of a Gold Crown

Exploring the principle of density and volume displacement, we will delve into a classic problem involving a gold crown to uncover misunderstandings and illustrate correct calculations. The problem presented involves a crown's density, mass, and volume displacement in a fluid, necessitating a careful approach to avoid common pitfalls.

Introduction to Density and Volume Displacement

A fundamental concept in physics and material science, the density of a substance is a measure of its mass per unit volume. The density of gold, for instance, is 19.3 g/cm3. This means 1 cm3 of gold has a mass of 19.3 grams. When an object is fully submerged in a fluid, it will displace an amount of the fluid equal to its own volume. This principle is famously attributed to Archimedes and is named after him.

The Given Scenario

A king's gold crown: The crown has a mass of 3.92 kg (3920 g). When fully immersed in pure water, it displaces 200 cm3 of water. We need to find the volume of the crown.

Many might initially conclude the crown's volume is 200 cm3, based on the displacement of water. However, this assumption is flawed due to a discrepancy in the provided density value.

Analysis and Calculation

Starting with the correct density of gold, 19.3 g/cm3, which is significantly different from the given value of 1969 g/cm3:

Correct Approach:

Density of gold: 19.3 g/cm3 Mass of crown: 3920 g Volume of crown mass / density Volume of crown 3920 g / 19.3 g/cm3 203.12 cm3

This calculation reveals the crown's actual volume is approximately 203.12 cm3. Thus, the correct volume of the crown is not 200 cm3, but rather 203.12 cm3.

Additional Insights

Plugging the numbers into the density formula, we can also determine the mass of the crown if it had a volume of 200 cm3. Using the density of gold (19.3 g/cm3):

Calculation:

V 200 cm3 D 19.3 g/cm3 Mass V × D Mass 200 cm3 × 19.3 g/cm3 Mass 3860 g (3.86 kg)

This exercise highlights the importance of using accurate units and the correct values in problem-solving. Incorrect or inconsistent units can lead to significant errors in calculations.

Conclusion

The volume of the gold crown is 203.12 cm3 based on the correct density of gold, 19.3 g/cm3. Understanding and applying the principles of density and volume displacement is crucial to solving such problems accurately. Always ensure to use correct and consistent units to avoid discrepancies in your calculations.

Keywords: density, volume displacement, Archimedes principle