12.4

Linear Thermal Expansion

| Normal Solids

Have you ever found the metal lid on a glass jar too tight to open? One solution is to run hot water over the lid, which loosens because the metal expands more than the glass does. To varying extents, most materials expand when heated and contract when cooled. The increase in any one dimension of a solid is called linear expansion, linear in the sense that the expansion occurs along a line. Figure 12-9 illustrates the linear expansion of a rod whose length is L0 when the temperature is T0. When the temperature increases to , the length becomes , where and are the magnitudes of the changes in temperature and length, respectively. Conversely, when the temperature decreases to , the length decreases to .

Figure 12-9 When the temperature of a rod is raised by , the length of the rod increases by .

For modest temperature changes, experiments show that the change in length is directly proportional to the change in temperature (). In addition, the change in length is proportional to the initial length of the rod, a fact that can be understood with the aid of Figure 12-10. Part a of the drawing shows two identical rods. Each rod has a length L0 and expands by when the temperature increases by . Part b shows the two heated rods combined into a single rod, for which the total expansion is the sum of the expansions of each part—namely, . Clearly, the amount of expansion doubles if the rod is twice as long to begin with. In other words, the change in length is directly proportional to the original length (). Equation 12.2 expresses the fact that is proportional to both L0 and () by using a proportionality constant , which is called the coefficient of linear expansion.

Figure 12-10 (a) Each of two identical rods expands by when heated. (b) When the rods are combined into a single rod of length 2L0, the “combined” rod expands by 2 .

| Linear Thermal Expansion of a Solid
| The length L0 of an object changes by an amount when its temperature changes by an amount :

| (12.2)

where is the coefficient of linear expansion.
Common Unit for the Coefficient of Linear Expansion:

Solving Equation 12.2 for shows that . Since the length units of and L0 algebraically cancel, the coefficient of linear expansion has the unit of (C°)−1 when the temperature difference is expressed in Celsius degrees (C°). Different materials with the same initial length expand and contract by different amounts as the temperature changes, so the value of depends on the nature of the material. Table 12-1 shows some typical values. Coefficients of linear expansion also vary somewhat depending on the range of temperatures involved, but the values in Table 12-1 are adequate approximations. Example 3 deals with a situation where a dramatic effect due to thermal expansion can be observed, even though the change in temperature is small.

Table 12-1 Coefficients of Thermal Expansion for Solids and Liquids*

| Coefficient of Thermal Expansion (C°)−1
Substance | Linear () | Volume ()
Solids
Aluminum |  |
Brass |  |
Concrete |  |
Copper |  |
Glass (common) |  |
Glass (Pyrex) |  |
Gold |  |
Iron or steel |  |
Lead |  |
Nickel |  |
Quartz (fused) |  |
Silver |  |
Liquids*
Benzene | — |
Carbon tetrachloride | — |
Ethyl alcohol | — |
Gasoline | — |
Mercury | — |
Methyl alcohol | — |
Water | — |

a) |  | The values for and pertain to a temperature near 20 °C.

b) |  | Since liquids do not have fixed shapes, the coefficient of linear expansion is not defined for them.

| Example 3 | | | Buckling of a Sidewalk

A concrete sidewalk is constructed between two buildings on a day when the temperature is 25 °C. The sidewalk consists of two slabs, each three meters in length and of negligible thickness (Figure 12-11a). As the temperature rises to 38 °C, the slabs expand, but no space is provided for thermal expansion. The buildings do not move, so the slabs buckle upward. Determine the vertical distance y in part b of the drawing.

Figure 12-11 (a) Two concrete slabs completely fill the space between the buildings. (b) When the temperature increases, each slab expands, causing the sidewalk to buckle.

Reasoning The expanded length of each slab is equal to its original length plus the change in length due to the rise in temperature. We know the original length, and Equation 12.2 can be used to find the change in length. Once the expanded length has been determined, the Pythagorean theorem can be employed to find the vertical distance y in Figure 12-11b.
Solution The change in temperature is , and the coefficient of linear expansion for concrete is given in Table 12-1. The change in length of each slab associated with this temperature change is

| (12.2)

The expanded length of each slab is, thus, 3.000 47 m. The vertical distance y can be obtained by applying the Pythagorean theorem to the right triangle in Figure 12-11b:

The buckling of a sidewalk is one consequence of not providing sufficient room for thermal expansion. To eliminate such problems, engineers incorporate expansion joints or spaces at intervals along bridge roadbeds, as Figure 12-12 shows.

Figure 12-12 An expansion joint in a bridge. (Richard Choy/Peter Arnold, Inc.)

Although Example 3 shows how thermal expansion can cause problems, there are also times when it can be useful. For instance, each year thousands of children are taken to emergency rooms suffering from burns caused by scalding tap water. Such accidents can be reduced with the aid of the antiscalding device shown in Figure 12-13. This device screws onto the end of a faucet and quickly shuts off the flow of water when it becomes too hot. As the water temperature rises, the actuator spring expands and pushes the plunger forward, shutting off the flow. When the water cools, the spring contracts and the water flow resumes.

Figure 12-13 An antiscalding device.

The physics of an antiscalding device.

| Thermal Stress

If the concrete slabs in Figure 12-11 had not buckled upward, they would have been subjected to immense forces from the buildings. The forces needed to prevent a solid object from expanding must be strong enough to counteract any change in length that would occur due to a change in temperature. Although the change in temperature may be small, the forces—and hence the stresses—can be enormous. They can, in fact, lead to serious structural damage. Example 4 illustrates just how large the stresses can be.

A N A L Y Z I N G M U L T I P L E - C O N C E P T P R O B L E M S

| Example 4 | | | The Stress on a Steel Beam

A steel beam is used in the roadbed of a bridge. The beam is mounted between two concrete supports when the temperature is 23 °C, with no room provided for thermal expansion (see Figure 12-14). What compressional stress must the concrete supports apply to each end of the beam, if they are to keep the beam from expanding when the temperature rises to 42 °C? Assume that the distance between the concrete supports does not change as the temperature rises.

Figure 12-14 A steel beam is mounted between concrete supports with no room provided for thermal expansion of the beam.

The physics of thermal stress.

Reasoning When the temperature rises by an amount , the natural tendency of the beam is to expand. If the beam were free to expand, it would lengthen by an amount (Equation 12.2). However, the concrete supports prevent this expansion from occurring by exerting a compressional force on each end of the beam. The magnitude F of this force depends on through the relation (Equation 10.17), where Y is Young’s modulus for steel and A is the cross-sectional area of the beam. According to the discussion in Section 10.8, the compressional stress is equal to the magnitude of the compressional force divided by the cross-sectional area, or Stress = F/A.
Knowns and Unknowns The data for this problem are listed in the table:

Description | Symbol | Value
Initial temperature | T0 | 23 °C
Final temperature | T | 42 °C
Unknown Variable |  |
Stress | – | ?

Modeling the Problem

| Step 1 Stress and Force
| The compressional stress is defined as the magnitude F of the compressional force divided by the cross-sectional area A of the beam (see Section 10.8), or

According to Equation 10.17, the magnitude of the compressional force that the concrete supports exert on each end of the steel beam is given by

where Y is Young’s modulus, is the change in length, and L0 is the original length of the beam. Substituting this expression for F into the definition of stress gives Equation 1 in the right column. Young’s modulus Y for steel is known (see Table 12-1), but we do not know either or L0. These two variables will be dealt with in Step 2.

| (1)

| Step 2 Linear Thermal Expansion
| If it were free to do so, the beam would have expanded by an amount (Equation 12.2), where is the coefficient of linear expansion. The change in temperature is the final temperature T minus the initial temperature T0, or . Thus, the beam would have expanded by an amount

In this expression the variables T and T0 are known, and can be obtained from Table 12-1. We substitute this relation for into Equation 1, as indicated in the right column.

Solution Algebraically combining the results of each step, we have
Note that the original length L0 of the beam is eliminated algebraically from this result. Taking the value of from Table 10-1 and from Table 12-1, we find that

Related Homework: Problems 18, 24, 90

| The Bimetallic Strip

A bimetallic strip is made from two thin strips of metal that have different coefficients of linear expansion, as Figure 12-15a shows. Often brass [] and steel [] are selected. The two pieces are welded or riveted together. When the bimetallic strip is heated, the brass, having the larger value of , expands more than the steel. Since the two metals are bonded together, the bimetallic strip bends into an arc as in part b, with the longer brass piece having a larger radius than the steel piece. When the strip is cooled, the bimetallic strip bends in the opposite direction, as in part c.

Figure 12-15 (a) A bimetallic strip and how it behaves when (b) heated and (c) cooled.

The physics of an automatic coffee maker. Bimetallic strips are frequently used as adjustable automatic switches in electrical appliances. Figure 12-16 shows an automatic coffee maker that turns off when the coffee is brewed to the selected strength. In part a, while the brewing cycle is on, electricity passes through the heating coil that heats the water. The electricity can flow because the contact mounted on the bimetallic strip touches the contact mounted on the “strength” adjustment knob, thus providing a continuous path for the electricity. When the bimetallic strip gets hot enough to bend away, as in part b of the drawing, the contacts separate. The electricity stops because it no longer has a continuous path along which to flow, and the brewing cycle is shut off. Turning the “strength” knob adjusts the brewing time by adjusting the distance through which the bimetallic strip must bend for the contact points to separate.

Figure 12-16 A bimetallic strip controls whether this coffee pot is (a) “on” (strip cool, straight) or (b) “off” (strip hot, bent).

| The Expansion of Holes

An interesting example of linear expansion occurs when there is a hole in a piece of solid material. We know that the material itself expands when heated. But what about the hole? Does it expand, contract, or remain the same? Conceptual Example 5 provides some insight into the answer to this question.

| Conceptual Example 5 | | | Do Holes Expand or Contract When the Temperature Increases?

Figure 12-17a shows eight square tiles that are arranged to form a square pattern with a hole in the center. If the tiles are heated, what happens to the size of the hole?

Figure 12-17 (a) The tiles are arranged to form a square pattern with a hole in the center. (b) When the tiles are heated, the pattern, including the hole in the center, expands. (c) The expanded hole is the same size as a heated tile.

Reasoning and Solution We can analyze this problem by disassembling the pattern into separate tiles, heating them, and then reassembling the pattern. Because each tile expands upon heating, it is evident from Figure 12-17b that the heated pattern expands and so does the hole in the center. In fact, if we had a ninth tile that was identical to and also heated like the others, it would fit exactly into the center hole, as Figure 12-17c indicates. Thus, not only does the hole in the pattern expand, but it also expands exactly as much as one of the tiles. Since the ninth tile is made of the same material as the others, we see that the hole expands just as if it were made of the material of the surrounding tiles. The thermal expansion of the hole and the surrounding material is analogous to a photographic enlargement; in both situations everything is enlarged, including holes.
Related Homework: Problems 12, 21

Instead of the separate tiles in Example 5, we could have used a square plate with a square hole in the center. The hole in the plate would have expanded just like the hole in the pattern of tiles. Furthermore, the same conclusion applies to a hole of any shape. Thus, it follows that a hole in a piece of solid material expands when heated and contracts when cooled, just as if it were filled with the material that surrounds it. If the hole is circular, the equation can be used to find the change in any linear dimension of the hole, such as its radius or diameter. Example 6 illustrates this type of linear expansion.

Problem solving insight

| Example 6 | | | A Heated Engagement Ring

A gold engagement ring has an inner diameter of and a temperature of 27 °C. The ring falls into a sink of hot water whose temperature is 49 °C. What is the change in the diameter of the hole in the ring? Reasoning The hole expands as if it were filled with gold, so the change in the diameter is given by , where is the coefficient of linear expansion for gold (Table 12-1), L0 is the original diameter, and is the change in temperature.
Solution The change in the ring’s diameter is

The previous two examples illustrate that holes expand like the surrounding material when heated. Therefore, holes in materials with larger coefficients of linear expansion expand more than those in materials with smaller coefficients of linear expansion. Conceptual Example 7 explores this aspect of thermal expansion.

| Conceptual Example 7 | | | Expanding Cylinders

Figure 12-18 shows a cross-sectional view of three cylinders, A, B, and C. Each is made from a different material: one is lead, one is brass, and one is steel. All three have the same temperature, and they barely fit inside each other. As the cylinders are heated to the same, but higher, temperature, cylinder C falls off, while cylinder A becomes tightly wedged to cylinder B. Which cylinder is made from which material?

Figure 12-18 Conceptual Example 7 discusses the arrangements of the three cylinders shown in cutaway views in parts a and b.

Reasoning and Solution We need to consider how the outer and inner diameters of each cylinder change as the temperature is raised. With respect to the inner diameter, we will be guided by the fact that a hole expands as if it were filled with the surrounding material. According to Table 12-1, lead has the greatest coefficient of linear expansion, followed by brass, and then by steel. These data indicate that the outer and inner diameters of the lead cylinder change the most, while those of the steel cylinder change the least.
Since the steel cylinder expands the least, it cannot be the outer one, for if it were, the greater expansion of the middle cylinder would prevent the steel cylinder from falling off. The steel cylinder also cannot be the inner one, because then the greater expansion of the middle cylinder would allow the steel cylinder to fall out, contrary to what is observed. The only place left for the steel cylinder is in the middle, which leads to the two possibilities in Figure 12-18. In part a, lead is on the outside and will fall off as the temperature is raised, since lead expands more than steel. On the other hand, the inner brass cylinder expands more than the steel that surrounds it and becomes tightly wedged, as observed. Thus, one possibility is A = brass, B = steel, and C = lead.
In part b of the drawing, brass is on the outside. As the temperature is raised, brass expands more than steel, so the outer cylinder will again fall off. The inner lead cylinder has the greatest expansion and will be wedged against the middle steel cylinder. A second possible answer, then, is A = lead, B = steel, and C = brass.
Related Homework: Conceptual Question 3

Check Your Understanding 2

A metal ball has a diameter that is slightly greater than the diameter of a hole that has been cut into a metal plate. The coefficient of linear thermal expansion for the metal from which the ball is made is greater than that for the metal of the plate. Which one or more of the following procedures can be used to make the ball pass through the hole? (a) Raise the temperatures of the ball and the plate by the same amount. (b) Lower the temperatures of the ball and the plate by the same amount. (c) Heat the ball and cool the plate. (d) Cool the ball and heat the plate. (The answer is given at the end of the book.)
Background: The coefficient of linear thermal expansion is the key concept here. The behavior of a hole in a piece of solid material that is heated or cooled is also important.
For similar questions (including calculational counterparts), consult Self-Assessment Test 12.1. This test is described at the end of Section 12.5.

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