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Chapter 17: Problem 21

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

Expert verified
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

01

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.
02

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.
03

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.
04

Relate Volumetric and Linear Expansion

The relationship between the volumetric and linear expansion coefficients is: \(\alpha = 3\beta\)
05

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: \(\Delta r_{sh} = r_{sh} \beta \Delta T\)
06

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 \(\Delta T\).

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Most popular questions from this chapter

Even though steel has a relatively low linear expansion coefficient \(\left(\alpha_{\text {steel }}=13 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\right),\) the expansion of steel railroad tracks can potentially create significant problems on very hot summer days. To accommodate for the thermal expansion, a gap is left between consecutive sections of the track. If each section is \(25.0 \mathrm{~m}\) long at \(20.0{ }^{\circ} \mathrm{C}\) and the gap between sections is \(10.0 \mathrm{~mm}\) wide, what is the highest temperature the tracks can take before the expansion creates compressive forces between sections?

In a thermometer manufacturing plant, a type of mercury thermometer is built at room temperature \(\left(20^{\circ} \mathrm{C}\right)\) to measure temperatures in the \(20^{\circ} \mathrm{C}\) to \(70^{\circ} \mathrm{C}\) range, with \(\mathrm{a}\) \(1-\mathrm{cm}^{3}\) spherical reservoir at the bottom and a \(0.5-\mathrm{mm}\) inner diameter expansion tube. The wall thickness of the reservoir and tube is negligible, and the \(20^{\circ} \mathrm{C}\) mark is at the junction between the spherical reservoir and the tube. The tubes and reservoirs are made of fused silica, a transparent glass form of \(\mathrm{SiO}_{2}\) that has a very low linear expansion coefficient \((\alpha=\) \(\left.0.4 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\right) .\) By mistake, the material used for one batch of thermometers was quartz, a transparent crystalline form of \(\mathrm{SiO}_{2}\) with a much higher linear expansion coefficient \(\left(\alpha=12.3 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\right) .\) Will the manufacturer have to scrap the batch, or will the thermometers work fine, within the expected uncertainty of \(5 \%\) in reading the temperature? The volume expansion coefficient of mercury is \(\beta=181 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\).

In order to create a tight fit between two metal parts, machinists sometimes make the interior part larger than the hole into which it will fit and then either cool the interior part or heat the exterior part until they fogether. Suppose an aluminum rod with diameter \(D_{1}\) (at \(\left.2.0 \cdot 10^{1}{ }^{\circ} \mathrm{C}\right)\) is to be fit into a hole in a brass plate that has a diameter \(D_{2}=10.000 \mathrm{~mm}\) (at \(\left.2.0 \cdot 10^{1}{ }^{\circ} \mathrm{C}\right) .\) The machinists can cool the rod to \(77.0 \mathrm{~K}\) by immersing it in liquid nitrogen. What is the largest possible diameter that the rod can have at \(2.0 \cdot 10^{1}{ }^{\circ} \mathrm{C}\) and just fit into the hole if the rod is cooled to \(77.0 \mathrm{~K}\) and the brass plate is left at \(2.0 \cdot 10^{1}{ }^{\circ} \mathrm{C} ?\) The linear expansion coefficients for aluminum and brass are \(22 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\) and \(19 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\), respectively.

For a class demonstration, your physics instructor uniformly heats a bimetallic strip that is held in a horizontal orientation. As a result, the bimetallic strip bends upward. This tells you that the coefficient of linear thermal expansion for metal T, on the top is _____ that of metal B, on the bottom. a) smaller than b) larger than c) equal to

On a hot summer day, a cubical swimming pool is filled to within \(1.0 \mathrm{~cm}\) of the top with water at \(21{ }^{\circ} \mathrm{C} .\) When the water warms to \(37^{\circ} \mathrm{C}\), the pool overflows. What is the depth of the pool?
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