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

The mass of the sphere is equal to the buoyant force divided by gravity, and the density is the mass divided by the volume of the material.

Step 1: Calculate the Volume of the Submerged Sphere

The sphere is floating half-submerged, so the volume of the liquid displaced is half the volume of the sphere. Calculate the outer volume of the sphere using the formula for the volume of a sphere: \( V = \frac{4}{3} \pi R^3 \). Here, the outer radius \( R = 9 \) cm, so the volume is \( V_{outer} = \frac{4}{3} \pi (9)^3 \). Then, divide it by 2 to find the submerged volume.

Step 2: Calculate the Buoyant Force

The buoyant force is equal to the weight of the displaced liquid. Use the formula \( F_b = \rho_{liquid} \cdot V_{submerged} \cdot g \), where \( \rho_{liquid} = 800 \text{ kg/m}^3 \), \( V_{submerged} \) is the submerged volume calculated in Step 1, and \( g = 9.8 \text{ m/s}^2 \). This force equals the weight of the sphere.

Step 3: Determine the Mass of the Sphere

Equate the buoyant force to the weight of the sphere (\( F_b = m_{sphere} \cdot g \)) to find the mass of the sphere: \( m_{sphere} = \frac{F_b}{g} \). From Step 2, \( F_b \) is known, and \( g \) is the acceleration due to gravity, so solve for \( m_{sphere} \).

Step 4: Calculate the Volume of the Material of the Sphere

Calculate the hollow (inner) volume using the radius \( r = 8 \) cm: \( V_{inner} = \frac{4}{3} \pi (8)^3 \). The volume of the material is the difference between the outer and inner volumes: \( V_{material} = V_{outer} - V_{inner} \).

Step 5: Calculate the Density of the Material

Density \( \rho = \frac{mass}{volume} \). Use the mass from Step 3 and the volume of the material from Step 4: \( \rho_{material} = \frac{m_{sphere}}{V_{material}} \).

Key Concepts

Density Calculation

Density is a fundamental property that describes how much mass is contained within a certain volume. In this exercise, the density of the sphere's material is what we need to find. The formula for density is:\[ \rho = \frac{mass}{volume} \]For the hollow sphere, we already know the mass from previous calculations.
Then, we need to calculate the volume of the material that makes up the sphere. The density of the material is crucial because it determines how the sphere behaves in the liquid.

• A higher density results in the object being more likely to sink.
• A lower density results in the object being more likely to float.

To find the density of the material, we calculate the difference in volume between the outer and inner spheres. Understanding the concept of density helps us grasp why some materials are more buoyant than others.

Volume of a Sphere

The volume of a sphere is a key piece of the puzzle in this exercise. Calculating the volume of spherical shapes can be essential in a variety of physics problems.The formula used to find the volume of a sphere is:\[ V = \frac{4}{3} \pi R^3 \]In this scenario, we have two radii to consider:- The outer radius of the sphere, which is 9 cm.- The inner radius of the hollow part, which is 8 cm.Understanding how to use these in the formula helps us find both the inner and outer volumes.
These calculations are necessary for determining the mass and density of the sphere.
Using this formula aids in determining the volume of any spherical object, making it an important tool in physics and geometry.

Mass Determination

Determining the mass of an object is fundamental in many scientific calculations, especially when dealing with buoyancy.In this exercise, the mass of the sphere is found by equating the buoyant force to gravitational force.The steps are:

• Calculate the buoyant force using the fluid's density and the sphere's submerged volume.
• Use the relationship \( F = m \cdot g \) to equate the buoyant force with mass.
• Solve for mass: \( m = \frac{F_b}{g} \)

By working through these steps, we determine how much matter is present in the sphere.
This process demonstrates the balance between gravitational forces and buoyant forces, which is essential in understanding why objects float or sink in liquids.