91Ó°ÊÓ

Archimedes' Principle is a key concept in understanding buoyancy. It states that an object submerged in a fluid experiences a buoyant force equal to the weight of the fluid displaced by the object. In simpler terms, when you put something in water, the water pushes back with a force.

This principle helps explain why objects float or sink. If the buoyant force is greater than or equal to the weight of the object, it floats. If it's less, the object sinks.

• Buoyant Force = Weight of Displaced Water
• The Force is directed upwards
• For an object to stay afloat, its weight must equal the buoyant force

In our exercise, the hollow glass shell barely floats, indicating the buoyant force perfectly balances the weight of the shell itself.
Recognizing this balance is essential in solving problems related to floating objects. It directs us to equate the volume of water displaced to the mass of the object to find parameters like the outer radius of the spherical shell.

Density is another fundamental concept in physics that helps us understand how much mass is packed into a given volume. Mathematically, it's expressed as \(\rho = \frac{m}{V}\), where \(\rho\) is density, \(m\) is mass, and \(V\) is volume.

Knowing the density of glass is crucial in our exercise. The density allows us to determine how much glass takes up space and contributes to the shell's weight.

• Glass density: \(2.60 \times 10^{3} \text{ kg/m}^3\)
• Determines how much space the mass of 1 kg occupies
• Helps find the volume of the material used in the shell

In our situation, the volume of the glass is calculated using its mass and density. This information is necessary to find the difference between the inner and outer radii of the shell, as it gives the physical space the glass occupies.

When dealing with spherical shells, especially a hollow one like ours, we need a mathematical toolset to correctly find the dimensions.

We must calculate both inner and outer radii, which involves understanding volumes and densities.

• Outer Volume equation: \(V_w = \frac{4}{3}\pi R_o^3\)
• Inner Volume adjustment for glass: \(V_g = \frac{4}{3}\pi (R_o^3 - R_i^3)\)

The problem is about using the density and mass of glass to find the glass's volume. Then, knowing the outer volume is derived from buoyancy, we deduce the shell's thickness by subtracting inner volume from outer volume.
These calculations form the backbone of our solution. The steps determine the precise dimensions required for a perfectly floating shell made entirely out of glass.