Chapter 9, Problem 78AP

The true weight of an object can be measured in a vacuum, where buoyant forces are absent. A measurement in air, however, is disturbed by buoyant forces. An object of volume V is weighed in air on an equal-arm balance with the use of counterweights of density ρ. Representing the density of air as ρ_air and the balance reading as ${\text{F}}_g^{\text{'}}$, show that the true weight F_g is

${{\displaystyle \text{F}}}_g={{\displaystyle \text{F}}}_g^{\text{'}}+\left(\text{v - }\frac{{{\displaystyle \text{F}}}_g^{\text{'}}}{\rho \text{g}}\right){\rho }_{air}\text{ g}$