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Chapter 11: Problem 73

A crown that is supposed to be made of solid gold is under suspicion. When the crown is weighed in air it has a weight of \(5.15 \mathrm{~N}\). When it is suspended from a digital balance and lowered into water, its apparent weight is measured to be \(4.88 \mathrm{~N}\). Given that the specific gravity of gold is \(19.3\), comment on the authenticity of the crown. SSM

Short Answer

Expert verified
Based on the calculations, the crown's density is less than the density of gold. Therefore, the crown is not made of solid gold.

Step by step solution

01

Calculate Density

The first step is to calculate the density of the crown. The density can be calculated by dividing its weight in air by the volume of water it displaces, which is equal to the difference in weight in air and weight in water. So, \(\rho = \frac{{\text{{Weight in air}}}}{{\text{{Weight in air}} - \text{{Weight in water}}}}\).
02

Calculate the Volume of water displaced

Next, calculate the volume of water displaced by the crown using the relation: \(\text{{volume}} = \frac{{\text{{force}}}}{{\text{{gravity}}}}\), where force here is the difference in weight in air and weight in water i.e., \((5.15 \mathrm{~N} - 4.88 \mathrm{~N})\) and gravity is \(9.8 \mathrm{m/s}^2\).
03

Calculate the true density of the crown

Now we substitute the values into the formula \(\rho = \frac{{\text{{Weight in air}}}}{{\text{{Weight in air}} - \text{{Weight in water}}}}\), plug in the known values and evaluate the expression to find the actual density of the crown.
04

Compare with the density of gold

Finally, compare the calculated density of the crown with the given specific gravity of gold, which is its density compared to water. Don't forget that the specific gravity needs to be multiplied by \(1000 \mathrm{kg/m}^3\) to convert it to a density comparable to our calculated result. If the densities are not equal, the crown is not made of pure gold.

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Key Concepts

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

Specific Gravity
Specific gravity is a measurement that compares the density of a substance to the density of water. Essentially, it's a unitless number because it's a ratio. This concept is vital in determining whether an object will sink or float when placed in a liquid. For instance, the specific gravity of gold is known to be 19.3.
This means gold is 19.3 times denser than water. In our exercise, to check if the crown is pure gold, we use the specific gravity of gold to see if the crown's density matches.
  • If the crown's density is vastly different from the specific gravity of gold, it might not be authentic.
  • This comparison provides a clue about whether the crown consists solely of gold or some other material mixed in.
Understanding specific gravity helps us identify materials based on their buoyant effects and density calculations.
Buoyant Force
Buoyant force is the upward force exerted by a fluid on a submerged object. According to Archimedes' Principle, this force equals the weight of the fluid that the object displaces. In simpler terms, if you drop something into water, the water pushes back up. This pushing force is the buoyant force.
The concept of buoyant force is crucial in our exercise when measuring the crown. When the crown is submerged in water, it appears to weigh less than in air. This difference in weight is the buoyant force acting on the crown.
  • Buoyant force allows us to determine how much water the crown displaces.
  • Understanding this helps in measuring the volume and, consequently, the density of the crown.
It's this understanding that ties into assessing whether the crown is made purely of gold or not.
Archimedes' Principle
Archimedes' Principle states that when an object is fully or partially submerged in a fluid, it experiences a buoyant force equal to the weight of the fluid that the object displaces. This principle is instrumental in calculating the density of objects through fluid displacement.
In our exercise, using Archimedes' Principle allows us to determine the crown's volume and density. By subtracting the crown's weight in water from its weight in air, we get the buoyant force. This buoyant force directly tells us how much water the crown displaces, and from this, we calculate volume.
  • Knowing the volume and the weight in air, we can find the density using density formulas.
  • Archimedes' Principle is the foundational concept behind this method of analysis and inspection.
This principle generally aids in authenticating the material composition of objects through simple fluid displacement tests.

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