- What is linear dilation?
- Formula of linear expansion and its coefficient
- Coefficient of linear expansion for various materials
- Worked Examples of Linear Dilation
- Example 1
- Solution
- Example 2
- Solution
- Example 3
- Solution
- Example 4
- Solution
- References
The linear expansion occurs when an object undergoes expansion due to a temperature variation, predominantly in one dimension. This is due to the characteristics of the material or its geometric shape.
For example, in a wire or a bar, when there is an increase in temperature, it is the length that suffers the greatest change due to thermal expansion.
Birds perched on wires. Source: Pixabay.
The cables on which the birds in the previous figure perch suffer a stretch when their temperature increases; instead, they contract when they cool. The same happens, for example, with the bars that form the rails of a railway.
What is linear dilation?
Graph of chemical bond energy versus interatomic distance. Source: self made.
In a solid material, the atoms maintain their relative positions more or less fixed around an equilibrium point. However, due to thermal agitation, they are always oscillating around it.
As the temperature increases, the thermal swing also increases, causing the middle swing positions to change. This is because the binding potential is not exactly parabolic and has asymmetry around the minimum.
Below is a figure that outlines the chemical bond energy as a function of the interatomic distance. It also shows the total energy of oscillation at two temperatures, and how the center of oscillation moves.
Formula of linear expansion and its coefficient
To measure linear expansion, we start with an initial length L and an initial temperature T, of the object whose expansion is to be measured.
Suppose that this object is a bar whose length is L and the cross-sectional dimensions are much less than L.
The object is first subjected to a temperature variation ΔT, such that the final temperature of the object once thermal equilibrium with the heat source has been established will be T '= T + ΔT.
During this process, the length of the object will also have changed to a new value L '= L + ΔL, where ΔL is the variation in length.
The coefficient of linear expansion α is defined as the quotient between the relative variation in length per unit variation in temperature. The following formula defines the coefficient of linear expansion α:
The dimensions of the coefficient of linear expansion are those of the inverse of temperature.
Temperature increases the length of tube-shaped solids. This is what is known as linear dilation. Source: lifeder.com
Coefficient of linear expansion for various materials
Next we will give a list of the coefficient of linear expansion for some typical materials and elements. The coefficient is calculated at normal atmospheric pressure based on an ambient temperature of 25 ° C; and its value is considered constant in a ΔT range of up to 100 ° C.
The unit of the coefficient of linear expansion will be (° C) -1.
- Steel: α = 12 ∙ 10 -6 (° C) -1
- Aluminum: α = 23 ∙ 10 -6 (° C) -1
- Gold: α = 14 ∙ 10 -6 (° C) -1
- Copper: α = 17 ∙ 10 -6 (° C) -1
- Brass: α = 18 ∙ 10 -6 (° C) -1
- Iron: α = 12 ∙ 10 -6 (° C) -1
- Glass: α = (7 to 9) ∙ 10 -6 (° C) -1
- Mercury: α = 60.4 ∙ 10 -6 (° C) -1
- Quartz: α = 0.4 ∙ 10 -6 (° C) -1
- Diamond: α = 1.2 ∙ 10 -6 (° C) -1
- Lead: α = 30 ∙ 10 -6 (° C) -1
- Oak wood: α = 54 ∙ 10 -6 (° C) -1
- PVC: α = 52 ∙ 10 -6 (° C) -1
- Carbon fiber: α = -0.8 ∙ 10 -6 (° C) -1
- Concrete: α = (8 to 12) ∙ 10 -6 (° C) -1
Most materials stretch with an increase in temperature. However, some special materials such as carbon fiber shrink with increasing temperature.
Worked Examples of Linear Dilation
Example 1
A copper cable is hung between two poles, and its length on a cool day at 20 ° C is 12 m. Find the value of its longitude on a hot day at 35 ° C.
Solution
Starting from the definition of the coefficient of linear expansion, and knowing that for copper this coefficient is: α = 17 ∙ 10 -6 (° C) -1
The copper cable undergoes an increase in its length, but this is only 3 mm. In other words, the cable goes from having 12,000 m to having 12,003 m.
Example 2
In a smithy, an aluminum bar comes out of the furnace at 800 degrees centigrade, measuring a length of 10.00 m. Once it cools to room temperature of 18 degrees Celsius, determine how long the bar will be.
Solution
In other words, the bar, once cold, will have a total length of:
9.83 m.
Example 3
A steel rivet has a diameter of 0.915 cm. A 0.910 cm hole is made on an aluminum plate. These are the initial diameters when the ambient temperature is 18 ° C.
To what minimum temperature must the plate be heated for the rivet to pass through the hole? The goal of this is that when the iron returns to room temperature, the rivet will be snug in the plate.
Figure for example 3. Source: own elaboration.
Solution
Although the plate is a surface, we are interested in the dilation of the diameter of the hole, which is a one-dimensional quantity.
Let us call D 0 the original diameter of the aluminum plate, and D the one it will have once heated.
Solving for the final temperature T, we have:
The result of the above operations is 257 ° C, which is the minimum temperature to which the plate has to be heated for the rivet to pass through the hole.
Example 4
The rivet and plate from the previous exercise are placed together in an oven. Determine what minimum oven temperature must be for the steel rivet to pass through the hole in the aluminum plate.
Solution
In this case, both the rivet and the hole will be dilated. But the coefficient of expansion of steel is α = 12 ∙ 10 -6 (° C) -1, while that of aluminum is α = 23 ∙ 10 -6 (° C) -1.
We then look for a final temperature T such that both diameters coincide.
If we call the rivet 1 and the aluminum plate 2, we find a final temperature T such that D 1 = D 2.
If we solve for the final temperature T, we are left with:
Next we put the corresponding values.
The conclusion is that the oven must be at least 520.5 ° C for the rivet to pass through the hole in the aluminum plate.
References
- Giancoli, D. 2006. Physics: Principles with Applications. Sixth Edition. Prentice Hall. 238–249.
- Bauer, W. 2011. Physics for Engineering and Sciences. Volume 1. Mac Graw Hill. 422-527.