Question

Brass weights are used to weigh an aluminum object on an analytical balance. The weighing is done one time in dry air and another time in humid air. What should the mass of the object be to produce a noticeable difference in the balance readings, provided the balance's sensitivity is \(m_{0}=0.100 \mathrm{mg} ?\) (The density of aluminum is \(\rho_{\mathrm{A}}=2.70 \cdot 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\); the density of brass is \(\rho_{11}=8.50 \cdot 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\), The density of the dry air is \(1.2285 \mathrm{~kg} / \mathrm{m}^{3}\), and the density of the humid air is \(1.2273 \mathrm{~kg} / \mathrm{m}^{3}\).).

Short answer

Based on the given densities, balance's sensitivity, and gravitational acceleration, the mass of the aluminum object that would produce a noticeable difference in the balance readings in dry air and humid air is approximately 224.0 kilograms.

Step-by-step solution

Calculate the noticeable difference in buoyancy force

First, we need to find the difference in buoyancy force that corresponds to the balance's sensitivity. The balance's sensitivity, \(m_0\), is given in mass units, so let's convert it to the force by multiplying it by the gravitational acceleration, \(g\): \(F_0 = m_0 \cdot g\)

Calculate the volume of aluminum to cause a noticeable difference

Next, we need to find the volume of aluminum, \(V_A\), that would cause a noticeable difference in buoyancy force. The buoyancy force is equal to the weight of the displaced fluid. The difference in buoyancy force caused by the difference in air density, \(\Delta \rho = \rho_{dry\,air} - \rho_{humid\,air}\), would be: \(\Delta F_b = V_A \cdot \Delta \rho \cdot g\) Now, we can solve for \(V_A\): \(V_A = \frac{\Delta F_b}{\Delta \rho \cdot g}\) By substituting the balance's sensitivity-related force we obtained in Step 1: \(V_A = \frac{F_0}{\Delta \rho \cdot g}\)

Calculate the mass of aluminum

Now that we have calculated the volume of aluminum, we can find the mass of the object that would produce a noticeable difference in the balance readings. We can use the density of aluminum, \(\rho_A\), to find the mass: \(m_A = \rho_A \cdot V_A\) By substituting the volume of aluminum we obtained in Step 2: \(m_A = \rho_A \cdot \frac{F_0}{\Delta \rho \cdot g}\)

Plug in the given values and solve for the mass of aluminum

Lastly, we can plug in the given values for the densities, balance's sensitivity, and gravitational acceleration (assuming \(g = 9.81 \frac{m}{s^2})\): \(m_A = 2.70 \cdot 10^3 \frac{kg}{m^3} \cdot \frac{0.100\cdot10^{-3} kg \cdot 9.81 \frac{m}{s^2}}{(1.2285 - 1.2273) \frac{kg}{m^3} \cdot 9.81 \frac{m}{s^2}}\) Calculating this expression: \(m_A \approx 224.0 kg\) Therefore, the mass of the aluminum object should be approximately 224.0 kg to produce a noticeable difference in the balance readings in dry air and humid air.

Key concepts

Buoyancy Force Calculation

Buoyancy is a fundamental concept in physics that explains why objects float or sink in a fluid. The buoyancy force is the upward force exerted by a fluid that opposes the weight of an object immersed in it. To calculate this force, we use the principle of buoyancy, also known as Archimedes' principle, which states that the buoyant force on an object in a fluid is equal to the weight of the fluid that the object displaces.

For the calculation of buoyancy force, we use the formula:
\[ F_b = V \times \rho_{fluid} \times g \]
where \( F_b \) is the buoyancy force, \( V \) is the volume of the displaced fluid (or the volume of the object), \( \rho_{fluid} \) is the density of the fluid, and \( g \) is the acceleration due to gravity (approximately \( 9.81 m/s^2 \) on Earth). It is crucial to express all quantities in compatible units to get a correct calculation of the buoyancy force. If the calculated buoyancy force is greater than the weight of the object, the object will rise to the surface and float. If it's less, the object will sink.

Density and Buoyancy

Density, often denoted by \( \rho \), plays a critical role in determining whether an object will float or sink in a fluid. It is a measure of how much mass is contained within a unit volume of a material, usually expressed in kilograms per cubic meter (\( kg/m^3 \)).

An object will float if its density is less than that of the fluid it is placed in because it will displace a volume of fluid equal to its own weight before it is fully submerged. Conversely, if the object's density is greater than that of the fluid, it will sink because it cannot displace enough fluid to balance its own weight. In the context of buoyancy:

\[ \text{If } \rho_{object} < \rho_{fluid}, \text{ the object floats.} \]
\[ \text{If } \rho_{object} > \rho_{fluid}, \text{ the object sinks.} \]

In our exercise, the concept of density is used to determine the mass of the aluminum object that would have a noticeable difference in buoyancy force when weighed in air with different densities. By understanding the relationship between density and buoyancy, we can predict and quantify the floating or sinking behavior of materials in various fluids.

Analytical Balance Sensitivity

The sensitivity of an analytical balance represents the smallest change in mass that the balance can detect. It is crucial in experiments where precise measurements of mass are necessary, such as in chemistry and physics labs. The sensitivity is denoted by \( m_0 \) and is expressed in mass units, like milligrams (mg).

In our exercise, the analytical balance has a sensitivity of \( 0.100 \text{ mg} \). This value is critical for detecting a noticeable difference in mass readings when weighing an object under different conditions, such as dry and humid air. The balance can detect changes in mass as small as 0.100 mg, which corresponds to a force difference (due to buoyancy changes) that must be calculated. By considering the balance's sensitivity, we ensure that the measured weight differences are significant and not just experimental anomalies or noise, providing us with reliable data for scientific analysis.

Therefore, the concepts learned through this exercise emphasize not only the theoretical understanding of buoyancy but also the practical implications of using sensitive equipment in scientific experiments to gather accurate measurements.