Question

Analytic balances are calibrated to give correct mass values for such items as steel objects of density \(\rho_{s}=\) \(8000.00 \mathrm{~kg} / \mathrm{m}^{3}\). The calibration compensates for the buoyant force arising because the measurements are made in air, of density \(\rho_{\mathrm{a}}=1.205 \mathrm{~kg} / \mathrm{m}^{3}\). What compensation must be made to measure the masses of objects of a different material, of density \(\rho\) ? Does the buoyant force of air matter?

Short answer

Answer: The compensation needed to measure the mass of objects with a different density is given by mc = m(1 - ρ_a(ρ - ρ_s)/(ρ_sρ)). The importance of the buoyant force of air depends on the density of the object being measured. For objects with a density close to steel, the buoyant force of air has a negligible impact. However, for objects with significantly lower or higher densities than steel, the buoyant force of air matters.

Step-by-step solution

Write down the buoyant force formula

The buoyant force formula is given by F = Vρg, where V is the volume of the submerged object, ρ is the density of the fluid (air), and g is the acceleration due to gravity.

Rewrite the buoyant force formula in terms of mass and densities

We know that the mass of an object is given by m = Vρ, where m is the mass, V is the volume, and ρ is the density of the object. We can rewrite the buoyant force formula as follows: F = m(ρ_g/ρ) where m is the mass, ρ_g is the density of the fluid, and ρ is the density of the object. Now, let's denote the mass measured by the balance without buoyancy compensation as m0 and mass after compensation as mc, we get: mc = m0 - m(ρ_a/ρ) Since the balance is calibrated for steel objects having density ρ_s, let's find the expression for mass without buoyancy compensation in terms of ρ_s. m0 = m(1 - ρ_a/ρ_s) Now, we need to find the compensation needed to measure the mass of objects with a different material density ρ.

Find the compensation (mass correction)

We can find the compensation needed by solving for mc in terms of ρ and ρ_s: mc = m(1 - ρ_a/ρ_s) - m(ρ_a/ρ) Factoring out m and simplifying, we get: mc = m(1 - ρ_a(ρ - ρ_s)/(ρ_sρ))

Determine the importance of the buoyant force of air

The buoyant force depends on the density of the fluid and the density of the object. Since the buoyant force formula involves subtracting the product of the densities, the importance of the buoyant force for materials having density close to steel (ρ ≈ ρ_s) will be very small, as the subtraction leads to a negligible difference. However, for objects with significantly lower or higher densities than steel, the buoyant force of air will matter.

Key concepts

Understanding Mass Measurement

When we talk about mass measurement, we're basing it on how much matter an object contains. Mass isn't affected by gravity, making it constant everywhere in the universe. However, when measuring mass on Earth, we often face the challenge of external forces like buoyancy.

To measure mass accurately using an analytical balance, it's important to account for buoyant forces exerted by air. This is especially crucial when using balances calibrated for specific materials, such as steel. Air, although not dense, exerts a slight buoyant force that can affect the outcome of a measurement. This happens because the air pushes upward on the object being measured, similar to how water buoys up a boat.

• Balances must be calibrated to consider these forces.
• Calibrating ensures reliable and accurate mass readings.

Knowing the mass precisely allows for more sophisticated scientific analyses and applications.

Exploring Density

Density is a measure of how much mass is squeezed into a given volume. It's represented by the formula \( \rho = \frac{m}{V} \), where \( m \) is mass and \( V \) is volume.

Objects with high density have tightly packed mass, while low-density objects are more spread out. Steel, with a density of \( 8000.00 \text{ kg/m}^3 \), is an example of a material with high density. The importance of density in measuring mass is tied to buoyancy. The buoyant force formula, \( F = V \rho g \), shows us how density interacts with volume and gravity.

• An object's density relative to air's density determines its buoyancy in air.
• Higher-density materials experience less buoyant effect.

This is why materials like steel are used for mass calibration, as their density results in minimal air buoyancy interference.

Importance of Calibration in Measurements

Calibration is the process of configuring an instrument to provide a result for a sample within an acceptable range. For analytical balances, it involves adjusting the settings to ensure they account for the buoyant force of air when measuring mass.

This means ensuring that even with the external influence of air, the measured value reflects the true mass of the object. Calibration is essential when dealing with diverse materials because each material's unique density will affect buoyancy differently. For the most accurate measurements:

• Balances should be regularly calibrated.
• Calibration should consider environmental factors like air density.

By doing this, any mass measurement becomes trustworthy and accurate, crucial for applications in scientific research and industrial processes.