The true weight of an object can be measured in a vacuum,...

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 $\rho$. Representing the density of air as $\rho_{\text {air }}$ and the balance reading as $F_{g}^{\prime}$, show that the true weight $F_{g}$ is $F_{g}=F_{g}^{\prime}+\left(V-\frac{F_{g}^{\prime}}{\rho g}\right) \rho_{\mathrm{air}} g$

Key Concepts

Buoyant Force

Buoyant force is the upward thrust exerted by a fluid on an immersed object, as described by Archimedes’ principle. This force equals the weight of the fluid displaced by the object and causes the apparent weight of the object measured in the fluid (like air) to be less than its true weight if measured in a vacuum.

True Weight vs. Apparent Weight

True weight refers to the gravitational force acting on an object when buoyant effects are absent, such as in a vacuum, whereas apparent weight is the reduced weight measured when the object is weighed in a fluid. The difference between these two is due to the buoyancy provided by the fluid, which partially offsets the gravitational force.

Density in Weight Measurement

Density is a measure of mass per unit volume and plays a critical role in weight measurements involving buoyancy. The densities of both the counterweights (used in the balance) and the surrounding fluid (air) influence the measurement, since the buoyant effect depends on the volume of displaced fluid, which is directly related to the object's volume and the fluid’s density.

Equal-Arm Balance

An equal-arm balance is a device designed to compare the mass of an object with known counterweights. In such a balance, any disturbance such as the buoyant force acting on the object and the counterweights must be taken into account, as they can lead to discrepancies between the measured (apparent) weight and the true weight of the object.

Correction for Buoyancy in Measurements

To determine the true weight from the apparent weight measured in air, it is essential to correct for the buoyant force acting on both the object and the counterweights. The correction involves adding a term that accounts for the volume of air displaced by the object minus the volume corresponding to the mass of the counterweights, each multiplied by the density of air and gravitational acceleration, thereby yielding the true weight.

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