Problem 39 - Fundamentals of Physics

Question

A hollow sphere of inner radius \(8.0 \mathrm{~cm}\) and outer radius \(9.0 \mathrm{~cm}\) floats half-submerged in a liquid of density \(800 \mathrm{~kg} / \mathrm{m}^{3}\). (a) What is the mass of the sphere? (b) Calculate the density of the material of which the sphere is made.

Step-by-Step Solution

Answer

(a) The mass of the sphere is approximately 1.22 kg. (b) The density of the material is approximately 1355.56 kg/m³.

Step 1: Understand the Problem

We have a hollow sphere that is half-submerged in a liquid. We need to find the mass and density of the sphere material. We will use principles of buoyancy and volume calculations.

Step 2: Calculate Volume of the Sphere

The volume of a hollow sphere is given by the difference of the volumes of the outer sphere and the inner sphere. The formula for the volume of a sphere is \( V = \frac{4}{3}\pi r^3 \). For the outer radius \( R = 9 \) cm and inner radius \( r = 8 \) cm, convert to meters first \( R = 0.09 \) m and \( r = 0.08 \) m.Calculate the volume of the outer sphere: \( V_{outer} = \frac{4}{3}\pi (0.09)^3 \).Calculate the volume of the inner sphere: \( V_{inner} = \frac{4}{3}\pi (0.08)^3 \).The volume of the hollow part: \( V_{hollow} = V_{outer} - V_{inner} \).

Step 3: Determine the Submerged Volume

Since the sphere floats half-submerged, the submerged volume \( V_{submerged} \) is half of the outer volume.\( V_{submerged} = \frac{1}{2} V_{outer} \).

Step 4: Use Buoyancy to Find the Mass

The buoyant force is equal to the weight of the liquid displaced: \( F_b = \rho_{liquid} \cdot V_{submerged} \cdot g \).Here, \( \rho_{liquid} = 800 \) kg/m³ and \( g = 9.8 \) m/s².Equate this to the weight of the sphere (\( m \cdot g \)), giving: \( m = \rho_{liquid} \cdot V_{submerged} \).

Step 5: Calculate the Density of the Material

Once the mass of the sphere \( m \) is known, use the hollow volume \( V_{hollow} \) to find the density \( \rho_{material} \) of the sphere's material.\( \rho_{material} = \frac{m}{V_{hollow}} \).

Step 6: Perform the Calculations

1. Calculate \( V_{outer} = \frac{4}{3}\pi (0.09)^3 \approx 0.00305 \) m³.2. Calculate \( V_{inner} = \frac{4}{3}\pi (0.08)^3 \approx 0.00215 \) m³.3. Find \( V_{hollow} = V_{outer} - V_{inner} \approx 0.0009 \) m³.4. Find \( V_{submerged} = \frac{1}{2} \times 0.00305 \approx 0.001525 \) m³.5. Mass, \( m = 800 \times 0.001525 \approx 1.22 \) kg.6. Density, \( \rho_{material} = \frac{1.22}{0.0009} \approx 1355.56 \) kg/m³.

Key Concepts

Density Calculation

When working with objects immersed in fluids, understanding density is crucial. Density, often symbolized as \( \rho \), indicates how much mass is packed within a given volume. It is calculated as:

• \( \rho = \frac{m}{V} \)

where \( m \) is mass and \( V \) is volume. Considering a hollow sphere, the challenge is to calculate the density of the material itself, as it involves knowing both the mass of the hollow structure and the volume that it actually occupies. Once you find the sphere's mass from buoyancy principles, you use the volume of the solid shell, known as the hollow volume, to determine its density. This density helps explain why objects float or sink in different fluids.

Volume of Hollow Sphere

The volume of any hollow object like a sphere is determined by finding the difference between the volume of the entire outer structure and the inner empty portion. For a sphere, this is given by:

• Outer volume: \( V_{outer} = \frac{4}{3}\pi R^3 \)
• Inner volume: \( V_{inner} = \frac{4}{3}\pi r^3 \)
• Hollow volume: \( V_{hollow} = V_{outer} - V_{inner} \)

First, you must adjust radii from centimeters to meters, as calculations in the SI unit system are more consistent. This exercise focused on using the above expressions to derive the hollow space volumes when outer and inner radii were provided. Such calculations are crucial because they represent the actual volume which defines the weight and buoyancy capacity of the sphere.

Floating Objects

Objects float in a fluid based on the principle of buoyancy, which states an object in a fluid experiences an upward force equal to the weight of the fluid displaced by the object. This principle can be expressed with the equation:

• \( F_b = \rho_{liquid} \times V_{submerged} \times g \)

where \( \rho_{liquid} \) is the fluid's density, \( V_{submerged} \) the volume submerged in the fluid, and \( g \) the acceleration due to gravity. For objects like the hollow sphere half-submerged in a liquid, it means the weight of the sphere equals half the weight of the fluid displaced. Calculating this submerged volume and its relation to the entire object allows for determining the sphere's mass and checking the validity of its buoyancy in diverse fluids. This understanding helps predict how different materials and objects will behave in various liquids.