Question

Assume the density of brass weights to be \(8.0 \mathrm{~g} / \mathrm{cm}^{3}\) and that of air to be \(0.0012 \mathrm{~g} / \mathrm{cm}^{3}\). What fractional error arises from neglecting the buoyancy of air in weighing an object of density \(3.4 \mathrm{~g} / \mathrm{cm}^{3}\) on a beam balance?

Short answer

The fractional error in the measurement of the weight of the object due to the neglect of the buoyant force from the air it displaces is \( \frac{density_{air}}{density_{object}} = \frac{0.0012 \: g/cm^3}{3.4 \: g/cm^3} = 0.00035\), or approximately 0.035 %. The small percentage indicates a negligible error due to the air buoyancy in most practical cases.

Step-by-step solution

Understanding the problem

The beam balance measures the gravitational force exerted on the object. When an object is in the air, it displaces some air and experiences a buoyant force upwards due to the displaced air. If this buoyant force is neglected, the force measured by the balance (weight of the object) will be more than the actual weight of the object. Thus, there will be a fractional error in the measurement.

Calculate the Buoyant Force

Let's denote the volume of the object as \(V\), so the mass of the air displaced by the object can be calculated by multiplying this volume with the density of the air. The buoyant force \(F_b\) is equal to the weight of this displaced air, which is the product of its mass and the gravitational acceleration \(g\), so, \(F_b = V \times density_{air} \times g\)

Calculate the Actual Weight

The actual weight of the object is the product of its volume, its density, and the gravitational acceleration, \(W = V \times density_{object} \times g\)

Determine the Fractional Error

The fractional error is the ratio of the buoyant force (force neglected) to the actual weight of the object. This is calculated as \(fractional\ error = \frac{F_b}{W}\). Substituting the values found in steps 2 and 3 will provide the answer.

Key concepts

Density Basics

Density is a fundamental concept that describes how much mass is contained in a given volume. It is often expressed in units such as grams per cubic centimeter (\(\mathrm{g}/\mathrm{cm}^3\)). In this exercise, we explore the densities of brass weights and air. Brass has a high density of \(8.0 \mathrm{~g}/\mathrm{cm}^3\), meaning it is quite compact, while air has a very low density of \(0.0012 \mathrm{~g}/\mathrm{cm}^3\).

Understanding density is essential when considering buoyancy. Buoyancy occurs because an object in a fluid (like air or water) is subject to a force equal to the weight of the fluid displaced by the object. This concept explains why, for example, a brass weight in air results in a measurable buoyant force despite the low density of air.

When analyzing problems involving buoyancy, consider the comparative densities. Objects with higher density than their surrounding fluid sink, while those with lower density float or rise. Here, since the object's density (\(3.4 \mathrm{~g}/\mathrm{cm}^3\)) is greater than that of air, it experiences an upward force from the air's buoyancy, impacting weight measurements.

Measuring with a Beam Balance

A beam balance is a classic device used to measure the weight of an object by balancing it against a set of known weights. It's a seesaw-like apparatus where weights on either side must equalize for the beam to balance.

When using a beam balance, the system measures the gravitational force acting on the objects placed on it. However, when considering buoyancy, an unaccounted force remains: the buoyant force. This force can cause the measured weight to differ from the object's true weight. This is because the object, when weighed in air, displaces some air and thus experiences an upward buoyant force.

In practical terms, even the slight buoyant effect of air can introduce errors. For precise measurements, the error due to buoyancy must be corrected or, at the very least, acknowledged. In high-precision activities, such as scientific research or engineering, it's essential to account for all forces acting on the object to ensure high accuracy in weight measurement.

The Role of Gravitational Force in Buoyancy

Gravitational force is the attraction between two masses. In the context of buoyancy and weight measurement, it is this force that the beam balance measures.

The gravitational force is calculated using the mass of an object and the acceleration due to gravity \(g\). The formula for gravitational force (weight \(W\)) is:
\[ W = m \times g \]

In the buoyancy problem, both the object and the displaced air are subject to gravitational force. The key detail is that the buoyant force, which is equal to the weight of the displaced air, acts in the opposite direction of the object's weight. This affects the reading on a beam balance since the actual force that the beam measures is slightly less due to this upward buoyant force.

Neglecting the buoyant force from the displaced air results in a fractional error. This error can be significant depending on the object's and air's densities involved. Understanding these interactions is crucial for interpreting any deviations in measurements and for making necessary corrections to ensure precise data collection.