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Chapter 15: Problem 46

Archimedes purportedly used his principle to verify that the king's crown was pure gold by weighing the crown submerged in water. Suppose the crown's actual weight was \(25.0 \mathrm{N}\). What would be its apparent weight if it were made of (a) pure gold and (b) \(75 \%\) gold and \(25 \%\) silver, by volume? The densities of gold, silver, and water are \(19.3 \mathrm{g} / \mathrm{cm}^{3}, 10.5 \mathrm{g} / \mathrm{cm}^{3},\) and \(1.00 \mathrm{g} / \mathrm{cm}^{3},\) respectively.

Short Answer

Expert verified
If the crown was made of pure gold, the apparent weight would be \(23.7N\). If the crown was 75% gold and 25% silver by volume, the apparent weight would be smaller due to the increased buoyant force from a lower density.

Step by step solution

01

Find the volume of the crown

The weight of the crown when it is outside of the water is equal to the gravitational force: \(F_{g}=m \cdot g\), where \(m\) is the mass and \(g\) is the gravitational acceleration. Therefore, the crown's mass is \(m= F_{g}/g = 25.0N/9.8m/s^2 = 2.551 kg\). To find the volume of the crown, use the formula \(V=m/ \rho\), where \( \rho\) is the density. In the first case, it is pure gold so, \(rho_{gold}= 19.3 g/cm^3 = 19.3 kg/(dm)^3 = 19.3 kg/L = 19.3*10^3 kg/m^3\). Therefore, the volume \(V_{crown} =m / \rho_{gold}= 2.551kg / (19.3*10^3 kg/m^3) = 0.000132 m^3\).
02

Calculate apparent weight for pure gold crown

The apparent weight of the crown when it is submerged in water is equal to the actual weight minus the buoyant force. The buoyant force is calculated by multiplying the volume of the object by the density of the fluid it is submerged in and gravity: \(F_{b}= \rho_{fluid} \cdot V \cdot g\). In this case, \(\rho_{fluid}\) is the density of water, which is \(1.00 g/cm^3 = 1.00 kg/L = 1.00*10^3 kg/m^3\). So, \(F_{b}=1.00*10^3 kg/m^3 \cdot 0.000132 m^3 \cdot 9.8 m/s^2 = 1.30N\). Therefore, the apparent weight \(F_{a} = F_{g} - F_{b} = 25.0N - 1.30N = 23.7N\).
03

Calculate apparent weight for mixed gold and silver crown

In the second case, the crown is not pure gold. It is composed of 75% gold and 25% silver by volume, this means that the effective density of crown \(\rho_{crown}\) is : \(0.75 * \rho_{gold} + 0.25 * \rho_{silver}\) = 0.75 * 19.3 kg/m^3 + 0.25 *10.5 kg/m^3 = 17.2 kg/m^3. We repeat the process from step 1 and 2. But this time using \(\rho_{crown}\). The resultant apparent weight will be smaller because with the decrease in density, the buoyant force increases leading to the decrease in apparent weight.

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