Understanding the Problem: Copper Piece with a Cavity

The question describes a piece of copper that has an empty space inside, called a cavity. We are given its mass when weighed in air and its apparent mass when weighed in water. We also know the density of the copper material. Our goal is to find the volume of the internal cavity.

When the copper piece is weighed in air, we get its actual mass. When it is weighed in water, it appears lighter due to the buoyant force exerted by the water. This difference in weight is equal to the buoyant force.

Applying Archimedes' Principle and Buoyancy

Archimedes' principle states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. The volume of the displaced fluid is equal to the volume of the submerged part of the object.

Given data:

• Density of copper ($\rho_{copper}$) = 8.8 g/cm³
• Mass in air ($m_{air}$) = 264 g
• Apparent mass in water ($m_{water}$) = 221 g
• Density of water ($\rho_{water}$) $\approx$ 1 g/cm³ (standard assumption for problems like this unless specified otherwise)

Step 1: Calculate the Buoyant Force (or Mass of Displaced Water)

The buoyant force causes the object to appear lighter in water. The difference in mass (or weight) is due to the buoyant force.

Mass difference = Mass in air - Apparent mass in water

\begin{equation*} \text{Mass difference} = 264 \text{ g} - 221 \text{ g} = 43 \text{ g} \end{equation*}

This mass difference represents the mass of the water displaced by the copper piece. The weight of this displaced water is the buoyant force.

Step 2: Calculate the Total Volume of the Copper Piece (including cavity)

The mass of the displaced water (43 g) occupies a volume equal to the total volume of the submerged object (the copper piece with the cavity). We can calculate this volume using the density of water.

\begin{equation*} V_{total} = \frac{\text{Mass of displaced water}}{\rho_{water}} \end{equation*}

\begin{equation*} V_{total} = \frac{43 \text{ g}}{1 \text{ g/cm}^3} = 43 \text{ cm}^3 \end{equation*}

This 43 cm³ is the total volume occupied by the copper piece, which includes the volume of the actual copper material and the volume of the internal cavity.

Step 3: Calculate the Volume of the Copper Material Only

We know the mass of the copper material (264 g) and the density of copper (8.8 g/cm³). We can calculate the volume of the solid copper material using the density formula ($\rho = m/V$).

\begin{equation*} V_{copper} = \frac{m_{air}}{\rho_{copper}} \end{equation*}

\begin{equation*} V_{copper} = \frac{264 \text{ g}}{8.8 \text{ g/cm}^3} \end{equation*}

To calculate this:

\begin{equation*} V_{copper} = \frac{2640}{88} \text{ cm}^3 = \frac{1320}{44} \text{ cm}^3 = \frac{660}{22} \text{ cm}^3 = \frac{330}{11} \text{ cm}^3 = 30 \text{ cm}^3 \end{equation*}

So, the volume of the actual copper material is 30 cm³.

Step 4: Calculate the Volume of the Cavity

The total volume of the piece is the sum of the volume of the copper material and the volume of the internal cavity.

\begin{equation*} V_{total} = V_{copper} + V_{cavity} \end{equation*}

We know $V_{total} = 43 \text{ cm}^3$ and $V_{copper} = 30 \text{ cm}^3$. We can now find the volume of the cavity.

\begin{equation*} V_{cavity} = V_{total} - V_{copper} \end{equation*}

\begin{equation*} V_{cavity} = 43 \text{ cm}^3 - 30 \text{ cm}^3 \end{equation*}

\begin{equation*} V_{cavity} = 13 \text{ cm}^3 \end{equation*}

The volume of the internal cavity is 13 cm³.

Let's summarize the volumes:

The total volume (43 cm³) displaces 43 g of water (since $\rho_{water} = 1$ g/cm³), resulting in a buoyant force equivalent to 43 g. This force makes the 264 g copper piece appear to weigh 264 g - 43 g = 221 g in water, which matches the given information.

Conclusion on Cavity Volume

Based on the calculations using the mass in air, mass in water, and copper density, the volume of the internal cavity within the copper piece is found to be 13 cm³.

Revision Table: Copper Cavity Volume Problem

Additional Information: Copper Density and Buoyancy

Density: Density is a fundamental property of a substance, defined as its mass per unit volume. For copper, the density is given as 8.8 g/cm³. This means that every cubic centimeter of solid copper has a mass of 8.8 grams. Knowing the density helps us determine the volume of a substance if we know its mass, or vice versa.

Buoyancy: Buoyancy is the upward force exerted by a fluid on a submerged or partially submerged object. This force is caused by the pressure difference between the top and bottom of the object. The pressure increases with depth, so the pressure at the bottom is greater than the pressure at the top, resulting in a net upward force. This force is what makes objects float or appear lighter in water.

In this problem, the presence of the cavity increases the total volume of the object compared to a solid piece of copper of the same mass. Since the buoyant force depends on the total submerged volume (the volume of displaced water), the object displaces more water than a solid copper piece of the same mass would. This larger buoyant force explains the measured weight in water.

Understanding how to apply density and buoyancy principles allows us to analyze objects that are not uniform in structure, like this copper piece with a cavity.