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. 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 $$ F_{\varepsilon}=F_{\varepsilon}^{\prime}+\left(V-\frac{F_{g}^{\prime}}{\rho g}\right) \rho_{\mathrm{air}} g $$

The true weight of an object can be measured in a vacuum, where buoyant forces are absent. 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
$$
F_{\varepsilon}=F_{\varepsilon}^{\prime}+\left(V-\frac{F_{g}^{\prime}}{\rho g}\right) \rho_{\mathrm{air}} g
$$

Key Concepts

Density and Counterweight Corrections

Density plays a crucial role in determining both the buoyant force (through the density of the surrounding fluid) and the calibration of measurement instruments like balances. In weighing systems using counterweights, the density of the counterweights must be taken into account to accurately relate the balance reading to the true weight of the object, requiring corrections that incorporate both the volume of the object and the density of the air.

True Weight

True weight is the gravitational force that an object experiences when measured in an environment free from buoyant effects, such as in a vacuum. It represents the actual mass of the object multiplied by the acceleration due to gravity, unaltered by the surrounding medium.

Apparent Weight

Apparent weight is the perceived or measured weight of an object when it is immersed in a fluid (like air). This measurement does not equal the true weight because the buoyant force — the upward force exerted by the displaced fluid — acts on the object, thereby reducing its reading on a balance.

Buoyant Force

The buoyant force is the upward force exerted by a fluid that opposes the weight of an immersed object. According to Archimedes’ principle, this force equals the weight of the fluid displaced by the object. In the context of weighing in air, the buoyant force reduces the apparent weight compared to the true weight.

Archimedes’ Principle

Archimedes’ principle is a fundamental concept in fluid mechanics stating that a body submerged in a fluid experiences an upward force equal to the weight of the fluid displaced by it. This principle is key to understanding the differences between true and apparent weight, particularly in the context of buoyancy corrections.

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