Calculating Cavity Volume of a Metallic Sphere using Buoyancy

This problem involves using the principles of buoyancy and density to determine the volume of an internal cavity within a metallic sphere. We are given the weight of the sphere in air and in water, as well as the density of the material the sphere is made from (excluding the cavity).

Understanding Buoyancy and Archimedes' Principle

When an object is submerged in a fluid, it experiences an upward force called the buoyant force. Archimedes' Principle states that this buoyant force is equal to the weight of the fluid displaced by the object.

The buoyant force can also be calculated as the difference between the object's weight in air and its weight when submerged in the fluid.

Given:

• Weight of sphere in air ($W_{air}$) = 40 g (This is equivalent to the mass of the sphere, $m_{sphere}$, assuming g=1 for simplicity with gram units)
• Weight of sphere in water ($W_{water}$) = 20 g
• Density of the metallic material ($\rho_{material}$) = 8 gm/cc
• Density of water ($\rho_{water}$) = 1 gm/cc (Standard value)

We need to find the volume of the internal cavity ($V_{cavity}$).

Step-by-Step Solution to Find Cavity Volume

Step 1: Calculate the Buoyant Force

The buoyant force ($BF$) acting on the sphere when it is submerged in water is the difference between its weight in air and its weight in water.

$$BF = W_{air} - W_{water}$$

$$BF = 40 \text{ g} - 20 \text{ g}$$

$$BF = 20 \text{ g}$$

Note: Since we are working with 'grams' as a measure of weight/mass in this context, the buoyant force calculated in grams is effectively the mass of water displaced.

Step 2: Determine the Volume of Water Displaced

According to Archimedes' Principle, the buoyant force is equal to the weight of the fluid displaced. The weight of the displaced water is the mass of the displaced water multiplied by gravitational acceleration. Since we calculated the buoyant force in grams (representing mass), the mass of displaced water is 20 g.

We know the density of water ($\rho_{water} = 1$ gm/cc). We can find the volume of water displaced ($V_{displaced}$) using the formula: mass = density × volume.

$$m_{displaced} = \rho_{water} \times V_{displaced}$$

$$20 \text{ g} = 1 \text{ gm/cc} \times V_{displaced}$$

$$V_{displaced} = \frac{20 \text{ g}}{1 \text{ gm/cc}}$$

$$V_{displaced} = 20 \text{ cc}$$

The volume of water displaced by the fully submerged sphere is equal to the total external volume of the sphere ($V_{total}$).

$$V_{total} = V_{displaced} = 20 \text{ cc}$$

Step 3: Calculate the Volume of the Solid Material

The total mass of the sphere is given by its weight in air, which is 40 g. This mass comes only from the solid metallic material, as the cavity is empty.

We are given the density of the metallic material ($\rho_{material} = 8$ gm/cc). We can find the volume of the solid material ($V_{solid}$) using the formula: mass = density × volume.

$$m_{sphere} = \rho_{material} \times V_{solid}$$

$$40 \text{ g} = 8 \text{ gm/cc} \times V_{solid}$$

$$V_{solid} = \frac{40 \text{ g}}{8 \text{ gm/cc}}$$

$$V_{solid} = 5 \text{ cc}$$

Step 4: Determine the Volume of the Cavity

The total external volume of the sphere ($V_{total}$) is the sum of the volume of the solid material ($V_{solid}$) and the volume of the internal cavity ($V_{cavity}$).

$$V_{total} = V_{solid} + V_{cavity}$$

We know $V_{total} = 20$ cc and $V_{solid} = 5$ cc. We can now solve for $V_{cavity}$.

$$20 \text{ cc} = 5 \text{ cc} + V_{cavity}$$

$$V_{cavity} = 20 \text{ cc} - 5 \text{ cc}$$

$$V_{cavity} = 15 \text{ cc}$$

Summary of Volumes

The volume of the internal cavity is 15 cc.

Revision Table: Key Concepts

Additional Information on Density and Buoyancy

Density: Density ($\rho$) is a fundamental property of a substance, defined as its mass ($m$) per unit volume ($V$). The formula is $\rho = m/V$. Different substances have different densities. For example, the density of water is about 1 gm/cc or 1000 kg/m³ at room temperature, while the density of iron is much higher, around 7.8 gm/cc.

Buoyancy: Buoyancy is the upward force that opposes the weight of a partially or fully immersed object in a fluid. It's caused by the pressure difference between the top and bottom of the object. The pressure in a fluid increases with depth, so the pressure on the bottom surface of an object is greater than the pressure on the top surface, resulting in a net upward force.

Archimedes' Principle in Detail: This principle is crucial for understanding floating and sinking. An object floats if the buoyant force is equal to its weight. It sinks if its weight is greater than the maximum buoyant force (which occurs when the object is fully submerged). The maximum buoyant force is the weight of the fluid whose volume is equal to the object's total external volume.

In this problem, the metallic sphere with a cavity is denser overall than water (since it sinks, having a positive weight in water). However, the cavity reduces the average density of the sphere compared to the solid material's density (8 gm/cc). By measuring the weight in air and water, we effectively determine the sphere's total volume (from buoyant force) and the volume of its solid material (from its mass and the material's density), allowing us to isolate the cavity's volume.