Example 14.5 Eureka! Archimedes supposedly was asked to determine whether a crow

Question

Example14.5Eureka! 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 read7.51N when the crown was in air and6.91N when it was in water. What should Archimedes have told the king? SOLVE IT ConceptualizeThe figure helps us imagine what is happening in this example. Because of the buoyant force, the scale reading is smaller in figure (b) than in figure (a).

CategorizeThis problem is an example of a case 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.

AnalyzeWhen the crown is suspended in air, the scale reads the true weightT1=Fg(neglecting the small buoyant force due to the surrounding air). When the crown is immersed in water, the buoyant force reduces the scale reading to anapparentweight ofT2=Fg-B. Apply the force equilibrium condition to the crown in water: ?F=B+T2-Fg= 0 Apply the force equilibrium condition to the crown in water: ?F=B+T2-Fg= 0 Solve forBand substitute the known values: B=Fg-T2 =7.51N -6.91N =0.60N Solve forBand substitute the known values: B=Fg-T2 =7.51N -6.91N =0.60N B=Fg-T2 =7.51N -6.91N =0.60N Because this buoyant force is equal in magnitude to the weight of the displaced water,?wgVw=0.60N, whereVwis the volume of the displaced water and?wis its density. Also, the volume of the crownVcis equal to the volume of the displaced water because the crown is completely submerged. Find the volume of the crown: Vc=Vw= 0.60N ?wg = 0.60N (1000 kg/m3)(9.80 m/s2) Find the density of the crown from the equation: ?c= mc Vc = mcg Vcg MASTER ITHINTS:GETTING STARTED|I'M STUCK! (a) When the crown is suspended in air, the scale reads its true weight becauseT1=Fg(the buoyancy of air is negligible). (b) When the crown is immersed in water, the buoyant forceBchanges the scale reading to a lower valueT2=Fg-B.

Explanation / Answer

A) Density of platinum=21.45 g/cm3 mg=.117*9.8=1.1466N Fg= m/D *1g/cm^3 *9.8 =.117/21.45 *9.8 =.05345 N T=mg-Fg=1.1466-.05345=1.09315 N B) T=0 T=mg+m1g-Fg1 =>mg+m1g=Fg1 we know that Fg1 =Fgplatinum+Fgstr Fg platinum=1000*Vplatinum*9.8 mg-Fgplatinum=1.09315N =>Fgstr-m1g=1.09315 =>1000*V*9.8-84*V*9.8=1.09315 =>V=1.09315/9.8*916 =>V=1.22*10^-4 m^3

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