Example 14.5 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 the figure. Suppose the scale read 7.85 N when the crown was in air and 6.90 N when it was in water. What should Archimedes have told the king? SOLVE IT Conceptualize The figure helps us imagine what is happening in this example. Because of the buoyant force, the scale reading is smaller in F. figure (b) than in figure (a). a b Categorize This problem is an (a) When the crown is suspended in air, the scale reads its true weight example of a case discussed earlier because T1 = F (the buoyancy of air is negligible). (b) When the crown is because the crown is completely immersed in water, the buoyant force B changes the scale reading to a lower value T2 = Fg - B. 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 71 = F (neglecting the small buoyant force due to the surrounding air). When the crown is immersed in water, the buoyant force B reduces the scale reading to an apparent weight of T2 = F. - B. Apply the particle in equilibrium model to [F = B+ T2- Fg = 0 the crown in water: Solve for B: B = Fg - T2 Because this buoyant force is equal in magnitude to the weight of the displaced water, B = Pwg disp,Analyze When the crown is suspended in air, the scale reads the true weight T1 = F9 (neglecting the small buoyant force due to the surrounding air). When the crown is immersed in water, the buoyant force B reduces the scale reading to an apparent weight of T2 = Fg B. Apply the particle in equilibrium model to 2F: B + T2 Fg = 0 the crown in water: Solve for B: B = Fg T2 Because this buoyant force is equal in magnitude to the weight of the displaced water, 8 = pygisp, where Vmsp is the volume of the displaced water and 9,, is its density. Also, the volume of the crown V, is equal to the volume of the displaced water because the crown is completely submerged, so B = nwch Find the density of the crown from the may "7:9 _ mcgw = mcgpw equation: _ Veg _ (B/pw) B F, T2 7.85 N (1 000 kg/m3) 7.85 N 6.90 N Substitute numerical values: pt = 8263.16 'y kg/m3 17:: Finalize From this table, we see that the density of gold is 19.3 X 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. m HINTS: GETTING STARTED I I'M STUCK! (a) In a similar experiment, if we want to buoy a 0.187-kg solid platinum object in water with an attached string from above, what is the tension required to support the piece of platinum? i N (b) If the object is now attached to a piece of Styrofoam, what volume of Styrofoam (density p5 = 88 kg/m3) should be used to have the Styrofoam and the metal floating just under the water surface? Submit