Question

Imagine that you place a cork measuring \(1.30 \mathrm{~cm} \times 5.50 \mathrm{~cm} \times\) \(3.00 \mathrm{~cm}\) in a pan of water and that on top of the cork you place a small cube of lead measuring \(1.15 \mathrm{~cm}\) on each edge. The density of cork is \(0.235 \mathrm{~g} / \mathrm{cm}^{3}\), and the density of lead is \(11.35 \mathrm{~g} / \mathrm{cm}^{3}\). Will the combination of cork plus lead float or sink?

Short answer

The combination of cork and lead will float because its density is less than water's density.

Step-by-step solution

Calculate Volume of Cork

The volume of the cork is calculated by multiplying its dimensions: \( V_{\text{cork}} = 1.30 \, \text{cm} \times 5.50 \, \text{cm} \times 3.00 \, \text{cm} = 21.45 \, \text{cm}^3 \).

Calculate Mass of Cork

Use the formula \( \text{mass} = \text{density} \times \text{volume} \) to calculate the mass of the cork. Given that the density of cork is \( 0.235 \, \text{g/cm}^3 \), the mass is \( m_{\text{cork}} = 0.235 \, \text{g/cm}^3 \times 21.45 \, \text{cm}^3 = 5.039 \, \text{g} \).

Calculate Volume of Lead Cube

The volume of the lead cube is computed using the formula for the volume of a cube: \( V_{\text{lead}} = (1.15 \, \text{cm})^3 = 1.52 \, \text{cm}^3 \).

Calculate Mass of Lead Cube

Calculate the mass of the lead using its density and volume: \( m_{\text{lead}} = 11.35 \, \text{g/cm}^3 \times 1.52 \, \text{cm}^3 = 17.252 \, \text{g} \).

Calculate Total Mass of Cork and Lead

The total mass of the system is the sum of the mass of the cork and the mass of the lead: \( m_{\text{total}} = 5.039 \, \text{g} + 17.252 \, \text{g} = 22.291 \, \text{g} \).

Calculate Total Volume of Cork and Lead

The total volume is the sum of the volumes of the individual components: \( V_{\text{total}} = 21.45 \, \text{cm}^3 + 1.52 \, \text{cm}^3 = 22.97 \, \text{cm}^3 \).

Calculate Density of Cork and Lead System

Calculate the density of the cork plus lead combination using \( \text{density} = \frac{\text{mass}}{\text{volume}} \): \( \rho_{\text{system}} = \frac{22.291 \, \text{g}}{22.97 \, \text{cm}^3} \approx 0.970 \, \text{g/cm}^3 \).

Determine Floating Condition

The density of water is \(1.00 \, \text{g/cm}^3\). Since the density of the cork plus lead system \(0.970 \, \text{g/cm}^3\) is less than \(1.00 \, \text{g/cm}^3\), the combination will float.

Key concepts

Density Calculation

Density is a measure of how much mass is contained in a given volume. It is calculated by dividing mass by volume, represented by the formula \( \text{density} = \frac{\text{mass}}{\text{volume}} \). Understanding density is crucial because it determines whether an object will float or sink in a fluid, like water.
In the context of our exercise with the cork and the lead cube, each material has its own density. The cork, which is less dense, has a density of \( 0.235 \, \text{g/cm}^3 \), while the lead is much denser at \( 11.35 \, \text{g/cm}^3 \).
To find the combined density of both materials when they are together, we perform the following steps:

• Calculate the total mass by adding the mass of the cork and the lead.
• Calculate the total volume by summing up the volume of the cork and lead.
• Use the combined mass and volume to find the density using the density formula.

By doing this, we find that the system's density is \( 0.970 \, \text{g/cm}^3 \), which will be important to determine the floatation ability in the next section.

Floatation

Floatation refers to the ability of an object to stay afloat in a fluid, such as water. The basic principle guiding this is that an object will float if its density is less than the density of the fluid it is placed in.
Water has a density of \( 1.00 \, \text{g/cm}^3 \). In our exercise, the calculated density of the cork and lead combination is \( 0.970 \, \text{g/cm}^3 \). Since this is lower than the density of water, it indicates that the combination will indeed float.
Understanding this principle helps explain why some objects that seem heavy might float, while lighter ones might sink. It's all about the relative densities:

• If the object's density is less than the fluid, it floats.
• If the object's density is more than the fluid, it sinks.

This concept ties directly to the mass and volume calculations that determine density, as these are the key variables that decide whether an object will float or sink.

Mass and Volume Calculations

Mass and volume are the foundational components for calculating density. Knowing how to accurately calculate these is essential.
Mass is the amount of matter in an object and is usually measured in grams (g) or kilograms (kg). Volume is the amount of space that the object occupies, typically measured in cubic centimeters (cm"), liters (L), or cubic meters (m").
In our exercise, we start by finding the volume of the objects:

• Cork: Calculated using its dimensions, \(1.30 \, \text{cm} \times 5.50 \, \text{cm} \times 3.00 \, \text{cm} = 21.45 \, \text{cm}^3\).
• Lead: As it is a cube, the volume is \((1.15 \, \text{cm})^3 = 1.52 \, \text{cm}^3\).

Next, we calculate the masses using their respective densities:

• Cork: Mass = Density \( \times \) Volume = \( 0.235 \, \text{g/cm}^3 \times 21.45 \, \text{cm}^3 = 5.039 \, \text{g}\).
• Lead: Mass = \( 11.35 \, \text{g/cm}^3 \times 1.52 \, \text{cm}^3 = 17.252 \, \text{g}\).

With these values, we can then find the total mass and total volume, paving the way for the density calculation discussed earlier.