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Density is a critical concept that helps us understand buoyancy and floatation. It tells us how much mass is contained within a volume. The density (\(\rho\)) of a material is given by the formula:
\[ \rho = \frac{m}{V} \]where

• \(m\) is the mass of the object, usually measured in grams (g).
• \(V\) is the volume, measured in cubic centimeters (cm³).

The exercise involves an iron shell with a density of \(7.87 \, \text{g/cm}^3\). This means that every cubic centimeter of iron has a mass of 7.87 grams. Understanding the density of the materials involved is key to understanding how and why objects float. It provides insight into the relationship between the object and the fluid it displaces. This concept helps us introduce the principle of buoyancy that we'll explore further.

The volume of a sphere is determined using a simple formula that takes into account its diameter. For any sphere with diameter \(d\), the volume \(V\) is given by:
\[ V = \frac{4}{3} \pi \left(\frac{d}{2}\right)^3 \]This formula is vital in the given exercise to calculate the external and internal volumes of the hollow sphere.
For example, the outer diameter is given as 60 cm, so the outer volume is calculated as:

• \( \frac{4}{3} \pi \times (30)^3 \)

The volume of a sphere not only helps in determining size but also in understanding how much space it occupies under water. This is linked to the concept of volume displacement, since the outer and inner spheres together create a hollow space. This unique feature makes understanding the volume crucial for determining why and how much of the sphere is submerged.

Floatation is explained by the principles of buoyancy. An object floats when the buoyant force exerted by the water equals the gravitational force downward. When objects are submerged, they displace an amount of water equal to the volume of the object submerged. The condition for floatation can be expressed as:
\[ \text{weight of object} = \text{weight of water displaced} \]In the exercise, this translates to:

• \( \rho_{\text{iron}} \times V_{\text{shell}} = \rho_{\text{water}} \times V_{\text{displaced}} \)

The density of water is typically 1 g/cm³, which simplifies the computation. This principle ensures that objects like the hollow iron shell float when their total weight equals the volume of water displaced. Therefore, understanding floatation helps solve the problem of finding the inner diameter by applying the buoyancy equations.

The relationship between mass, volume, and displacement is a core idea in fluid dynamics, especially in determining an object's ability to float. In the exercise, the mass of the iron shell is calculated using the formula:
\[ \text{mass} = \rho_{\text{iron}} \times (V_{\text{outer}} - V_{\text{inner}}) \]This mass represents the actual weight supported by the water through displacement. Since the exercise assumes complete submersion, the volume displaced equals the outer volume.
This is important because:

• The weight of the displaced water, determined by the outer volume, indicates how much buoyancy force is exerted.
• The difference in the outer and inner volume helps ascertain the internal space, which is critical for finding the shell's true empty space.

By understanding these relationships, we can determine how hollow structures behave in a fluid, and in turn solve for the inner diameter by equating these volumes and masses.