Question

A solid cylinder and a cylindrical shell, of identical radius and length and made of the same material, experience the same temperature increase \(\Delta T .\) Which of the two will expand to a larger outer radius?

Short answer

Answer: Both the solid cylinder and the cylindrical shell will expand to the same larger outer radius when they experience the same temperature increase.

Step-by-step solution

Calculate Volume Change

Firstly, let's calculate the volume change of each cylinder with respect to the temperature difference. Since both cylinders are made from the same material, they will both have the same coefficient of volumetric expansion alpha (\(\alpha\)), which relates volume change and temperature change as follows: \(\Delta V = V_{0} \alpha \Delta T\) Here, \(V_{0}\) is the initial volume of the cylinder, \(\Delta V\) is the change in volume, and \(\Delta T\) is the temperature difference.

Find the Volume Relationship of Both Cylinders

We know that the solid cylinder and the cylindrical shell have the same radius and length, so let's define the volume of the solid cylinder as \(V_{s}\) and the volume of the cylindrical shell as \(V_{sh}\). As the cylindrical shell can be imagined as the result of subtracting a smaller internal cylinder from the solid cylinder with the same height, we can write: \(V_{sh} = V_{s} - V_{i}\), where \(V_{i}\) is the volume of the internal cylinder.

Calculate the Linear Expansion

The linear expansion, which relates the initial size of any dimension to the size after a temperature change, can be described as: \(\Delta L = L_{0} \beta \Delta T\) Here, \(L_0\) is the initial length of the object, \(\Delta L\) is the change in length, and \(\beta\) is the linear expansion coefficient.

Relate Volumetric and Linear Expansion

The relationship between the volumetric and linear expansion coefficients is: \(\alpha = 3\beta\)

Calculate the Change in Outer Radius

Now, we will calculate the change in outer radius for both cylinders. Let's denote the initial radius of the solid cylinder and the cylindrical shell as \(r_{s}\) and denote the change in the outer radius as \(\Delta r_{s}\). Then: \(\Delta r_{s} = r_{s} \beta \Delta T\) For the cylindrical shell, let's denote the initial outer radius as \(r_{sh}\) and the initial inner radius as \(r_{i}\). Then the change in the outer radius for the cylindrical shell can be written as: \(\Delta r_{sh} = r_{sh} \beta \Delta T\)

Compare the Change in Outer Radius

Since both the solid cylinder and the cylindrical shell have the same initial dimensions and coefficients of linear expansion, we can observe the change in the outer radius directly: \(\Delta r_{s} = r_{s} \beta \Delta T = r_{sh} \beta \Delta T = \Delta r_{sh}\) Since the change in the outer radius \(\Delta r_{s}\) is equal to the change in the outer radius \(\Delta r_{sh}\), we can conclude that both the solid cylinder and the cylindrical shell will expand to the same larger outer radius when they experience the same temperature increase \(\Delta T\).

Key concepts

Coefficient of Volumetric Expansion

The coefficient of volumetric expansion (\(\alpha\)) is an essential concept in understanding how materials respond to temperature changes. When a substance is heated, its particles move more vigorously, causing the material to expand in volume. The coefficient of volumetric expansion describes how much a given volume of a material expands per degree of temperature increase. This is important when considering materials in construction, design, and various engineering fields.
The formula that relates volume change to temperature change is given by:\[\Delta V = V_{0} \alpha \Delta T\]

• \(\Delta V\) is the change in volume.
• \(V_{0}\) is the original volume.
• \(\Delta T\) is the temperature change.
• \(\alpha\) is the coefficient of volumetric expansion.

Understanding this concept helps us anticipate how an object will behave under varying temperatures, which is crucial for problem-solving in thermal dynamics.

Linear Expansion

Linear expansion is another way to describe how materials respond to temperature changes, but it focuses on the change in length rather than volume. When an object is heated, its length increases, and this change is governed by the linear expansion coefficient (\(\beta\)). The relationship between the original length and its change in length with a temperature change is:\[\Delta L = L_{0} \beta \Delta T\]

• \(\Delta L\) is the change in length.
• \(L_0\) is the initial length.
• \(\Delta T\) is the temperature change.
• \(\beta\) is the coefficient of linear expansion.

A useful relation between the volumetric and linear coefficients is given by \(\alpha = 3\beta\), indicating that the volume expansion is three times the linear expansion when changes are uniform in all three dimensions. This principle helps to predict dimensional changes efficiently.

Solid Cylinder

A solid cylinder is a common geometric shape characterized by its circular base and uniform cross-section along its length. Understanding the effects of thermal expansion in a solid cylinder is crucial in applications such as pistons in engines and rollers in machinery. When a solid cylinder experiences heat, it expands uniformly in volume due to its material's coefficient of thermal expansion.
The expansion can be viewed in terms of volume and linear changes:

• In volume: due to the coefficient of volumetric expansion, expressed as \(\Delta V = V_{0} \alpha \Delta T\).
• In length and radius: governed separately by linear expansion, described with \(\Delta L = L_{0} \beta \Delta T\) and \(\Delta r = r \beta \Delta T\) for any radius \(r\).

Knowing these factors allows efficient design and management to prevent failure due to unwanted stress or deformation from temperature variations.

Cylindrical Shell

A cylindrical shell is essentially a hollow cylinder, which can be visualized as a solid cylinder with a smaller cylinder removed from its center, having both an inner and outer radius. It's often used in pipelines, barrels, and structural components. When a cylindrical shell undergoes thermal expansion, both its outer and inner surfaces expand similarly due to the uniform material properties.
For a cylindrical shell, we calculate expansion similarly to the solid cylinder but pay attention to both radii:

• The change in the outer radius, \(\Delta r_{sh}\), follows \(\Delta r_{sh} = r_{sh} \beta \Delta T\).
• The inner radius changes likewise, and its effects are considered in structural analysis.

With both the solid cylinder and the cylindrical shell experiencing the same temperature increase and material properties, their outer radii expand equally, which is a practical consideration in uniform material design, ensuring stability and performance.