Question 14.5: Eureka! Archimedes supposedly was asked to determine whether...
Eureka!
Archimedes supposedly was asked to determine whether a crown made for the king consisted of pure gold. According to legend, he solved this problem by weighing the crown first in air and then in water as shown in Figure 14.11. Suppose the scale read 7.84 N when the crown was in air and 6.84 N when it was in water. What should Archimedes have told the king?
Conceptualize Figure 14.11 helps us imagine what is happening in this example. Because of the upward buoyant force on the crown, the scale reading is smaller in Figure 14.11b than in Figure 14.11a.
Categorize This problem is an example of Case 1 discussed earlier because the crown is completely submerged. The scale reading is a measure of one of the forces on the crown, and the crown is stationary. Therefore, we can categorize the crown as a particle in equilibrium.
Analyze When the crown is suspended in air, the scale reads the true weight T_1=F_g (neglecting the small buoyant force due to the surrounding air). When the crown is immersed in water, the buoyant force \overrightarrow{B} due to the water reduces the scale reading to an apparent weight of T_2=F_g-B.
Apply the particle in equilibrium model to the crown in water:
\sum F=B+T_2-F_g=0Solve for B:
B=F_g-T_2=m_c g-T_2Because this buoyant force is equal in magnitude to the weight of the displaced water, B=\rho_w g V_{\text {disp }}, where V_{\text {disp }} is the volume of the displaced water and \rho_w is its density. Also, the volume of the crown V_c is equal to the volume of the displaced water because the crown is completely submerged, so B=\rho_w g V_c.
Find the density of the crown from Equation 1.1:
\rho \equiv \frac{m}{V} (1.1)
\rho_c=\frac{m_c}{V_c}=\frac{m_c g}{V_c g}=\frac{m_c g}{\left(B / \rho_w\right)}=\frac{m_c g \rho_w}{B}=\frac{m_c g \rho_w}{F_g-T_2}=\frac{m_c g \rho_w}{m_c g-T_2}Substitute numerical values:
\rho_c=\frac{(7.84 N)\left(1000 kg / m^3\right)}{7.84 N-6.84 N}=7.84 \times 10^3 kg / m^3Finalize From Table 14.1, we see that the density of gold is 19.3 × 10³ kg/m³. Therefore, Archimedes should have reported that the king had been cheated. Either the crown was hollow, or it was not made of pure gold.
| Table 14.1 Densities of Some Common Substances at Standard Temperature (0°C) and Pressure (Atmospheric) |
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| Substance | ρ (kg/m³) | Substance | ρ (kg/m³) |
| Air | 1.29 | Iron | 7.86 × 10³ |
| Air (at 20°C and | Lead | 11.3 × 10³ | |
| atmospheric pressure) | 1.20 | Mercury | 13.6 × 10³ |
| Aluminum | 2.70 × 10³ | Nitrogen gas | 1.25 |
| Benzene | 0.879 × 10³ | Oak | 0.710 × 10³ |
| Brass | 8.4 × 10³ | Osmium | 22.6 × 10³ |
| Copper | 8.92 × 10³ | Oxygen gas | 1.43 |
| Ethyl alcohol | 0.806 × 10³ | Pine | 0.373× 10³ |
| Fresh water | 1.00 × 10³ | Platinum | 21.4 × 10³ |
| Glycerin | 1.26 × 10³ | Seawater | 1.03 × 10³ |
| Gold | 19.3 × 10³ | Silver | 10.5 × 10³ |
| Helium gas | 1.79 \times 10^{-1} | Tin | 7.30 × 10³ |
| Hydrogen gas | 8.99 \times 10^{-2} | Uranium | 19.1 × 10³ |
| Ice | 0.917 × 10³ | ||