Calculating Area of Thermal Expansion for Non-Isotropic Materials

In summary, a non-isotropic material with lengths l1=6.1 cm and l2=6.0 cm experiences a fractional increase in length along different directions. The coefficient of expansion in the direction of l1 is \alpha1= 10×10-5 K-1 and in the direction of l2 is \alpha2= 40×10-5K-1. When heated, the rectangular plate becomes a perfect square. The area of the square is 38 cm2. However, to solve for the temperature, the equations ΔL=\alphaL0T, Lx=Lx0+Lx0\alphaΔT, Ly=Ly0+Ly0\alphaΔT,
  • #1
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Homework Statement


A non-isotropic material is heated, fractional increase in length is different along different directions. With length l1=6.1 cm and l2=6.0 cm. In direction of l1, the coefficient expansion is [itex]\alpha[/itex]1= 10×10-5 K-1 and direction l2 is [itex]\alpha[/itex]2= 40×10-5K-1. The plate is heated until it becomes a perfect square. What is the area of the square? (The answer is 38 cm2) I just don't know how use the equations without a given temperature.


Homework Equations


ΔL=[itex]\alpha[/itex]L0T
Lx=Lx0+Lx0[itex]\alpha[/itex]ΔT
Ly=Ly0+Ly0[itex]\alpha[/itex]ΔT
A0=Lx0Ly0
A=LxLy

The Attempt at a Solution


Not sure what to do with the equations without the temperature.
 
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  • #2
You need to find the temperature from the condition that the rectangular becomes square.
 

1. What is the definition of "Area of Thermal Expansion"?

The "Area of Thermal Expansion" is a measure of how much a material's surface area changes when it is heated or cooled. It is typically expressed as a percentage change in area per degree of temperature change.

2. How is the area of thermal expansion calculated?

The area of thermal expansion can be calculated by multiplying the original surface area by the coefficient of thermal expansion (CTE) and the change in temperature. The formula is: ΔA = A * CTE * ΔT, where ΔA is the change in area, A is the original area, CTE is the coefficient of thermal expansion, and ΔT is the change in temperature.

3. What factors affect the area of thermal expansion?

The area of thermal expansion is influenced by various factors including the type of material, its shape and dimensions, and the temperature change. Additionally, factors such as impurities, crystal structure, and internal stresses can also affect the expansion of a material.

4. Why is the area of thermal expansion important?

The area of thermal expansion is important because it can impact the performance and functionality of materials in various applications. For example, in construction, the expansion and contraction of materials due to temperature changes can cause cracks and structural damage. In engineering, the precise calculation of thermal expansion is crucial for the design and functioning of machines and structures.

5. How does thermal expansion differ from material to material?

The thermal expansion of a material can vary significantly depending on its composition and properties. For instance, metals generally have a higher coefficient of thermal expansion compared to ceramics. Additionally, the crystal structure and arrangement of atoms in a material can also affect its thermal expansion behavior. Therefore, it is essential to consider the specific characteristics of each material when calculating its area of thermal expansion.

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