when heating a solid the length increases from 5.2 cm to 5.8 cm in 10 calculate the linear expansion coefficient?
Answers
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Use the equation for linear thermal expansion \boldsymbol{\Delta{L}=\alpha{L}\Delta{T}} to calculate the change in length, ΔL, Use the coefficient of linear expansion, \boldsymbol{\alpha}, for steel from Table 2, and note that the change in temperature, ΔT , is 55 oC.
Solution
Plug all of the known values into the equation to solve for ΔL.
\Delta{L} = \alpha{L}\Delta{T}\:= \frac {12\times10^{-6}} {^{\circ} {C} } ( 1275{ m}) ( 55^{\circ}{C}) = 0.84 { m.}}
Discussion
Although not large compared with the length of the bridge, this change in length is observable. It is generally spread over many expansion joints so that the expansion at each joint is small.
Thermal Expansion in Two and Three Dimensions
Objects expand in all dimensions, as illustrated in Figure 2. That is, their areas and volumes, as well as their lengths, increase with temperature. Holes also get larger with temperature. If you cut a hole in a metal plate, the remaining material will expand exactly as it would if the plug was still in place. The plug would get bigger, and so the hole must get bigger too. (Think of the ring of neighbouring atoms or molecules on the wall of the hole as pushing each other farther apart as temperature increases. Obviously, the ring of neighbours must get slightly larger, so the hole gets slightly larger).
THERMAL EXPANSION IN TWO DIMENSIONS
For small temperature changes, the change in area ΔA is given by
\boldsymbol{\Delta{A}=2\alpha{A}\Delta{T}},
where ΔA is the change in area A , and ΔT is the change in temperature, and α is the coefficient of linear expansion, which varies slightly with temperature.
Example 2: Calculating Thermal Expansion: Gas vs. Gas Tank
Suppose your 60.0-L (15.9-gal) steel gasoline tank is full of gas, so both the tank and the gasoline have a temperature of\boldsymbol{15.0^{\circ}\textbf{C}}.How much gasoline has spilled by the time they warm to\boldsymbol{35.0^{\circ}\textbf{C}}?
Strategy
The tank and gasoline increase in volume, but the gasoline increases more, so the amount spilled is the difference in their volume changes. (The gasoline tank can be treated as solid steel.) We can use the equation for volume expansion to calculate the change in volume of the gasoline and of the tank.
Solution
1. Use the equation for volume expansion to calculate the increase in volume of the steel tank:
\boldsymbol{\Delta{V}_{\textbf{s}}=\beta_{\textbf{s}}V_{\textbf{s}}\Delta{T}}.
2. The increase in volume of the gasoline is given by this equation:
\boldsymbol{\Delta{V}_{\textbf{gas}}=\beta_{\textbf{gas}}V_{\textbf{gas}}\Delta{T}}.
3. Find the difference in volume to determine the amount spilled as
\boldsymbol{V_{\textbf{spill}}=\Delta{V}_{\textbf{gas}}-\Delta{V}_{\textbf{s}}}.
Alternatively, we can combine these three equations into a single equation. (Note that the original volumes are equal.)
\begin{array}{lcl} \boldsymbol{V_{\textbf{spill}}} & \boldsymbol{=} & \boldsymbol{(\beta_{\textbf{gas}}-\beta_{\textbf{s}})V\Delta{T}} \\ {} & \boldsymbol{=} & \boldsymbol{[(950-35)\times10^{-6}\textbf{/}^{\circ}\textbf{C}](60.0\textbf{ L})(20.0^{\circ}\textbf{C})} \\ {} & \boldsymbol{=} & \boldsymbol{1.10\textbf{ L.}} \end{array}
Discussion
A cracked asphalt road with a pothole.
Figure 5. Thermal stress contributes to the formation of potholes. (credit: Editor5807, Wikimedia Commons)
Metal is regularly used in the human body for hip and knee implants. Most implants need to be replaced over time because, among other things, metal does not bond with bone. Researchers are trying to find better metal coatings that would allow metal-to-bone bonding. One challenge is to find a coating that has an expansion coefficient similar to that of metal. If the expansion coefficients are too different, the thermal stresses during the manufacturing process lead to cracks at the coating-metal interface.
Another example of thermal stress is found in the mouth. Dental fillings can expand differently from tooth enamel. It can give pain when eating ice cream or having a hot drink. Cracks might occur in the filling. Metal fillings (gold, silver, etc.) are being replaced by composite fillings (porcelain), which have smaller coefficients of expansion, and are closer to those of teeth.