Derive the relation between alpha beta and gama
Answers
Answer:
As the temperature increases, the volume of the material also increases. This is known as thermal expansion. It can also be explained as the fractional change in the length or volume per unit change in the temperature.
The relation between alpha, beta, and gamma is given in the form of a ratio and the ratio is 1:2:3 and can be expressed as:
alpha=fracbeta2=fracgamma3
Following is the relation between the three:
L = L (1 + α.ΔT)
Where, α is the coefficient of linear expansion
A = A (1 + β.ΔT)
Where, β is the coefficient of aerial expansion
V = V (1 + γ.ΔT)
Where, γ is the coefficient of cubical expansion
V = V + γV.ΔT
V = V (1 + γ.ΔT)
L3 = L3 (1 + α.ΔT)3
L3 = L3 (1 + 3α.ΔT + 3α2.ΔT2 + α3.ΔT3)
L3 = L3 (1 + 3α.ΔT)
Since 3α2.ΔT2 and α3.ΔT3 are smaller than 1, we are not considering them.
L3 = L3 (1 + 3α.ΔT)
V = L3 (1 + 3α.ΔT)
V (1 + γ.ΔT) = V (1 + 3α.ΔT)
1 + γ.ΔT = 1 + 3α.ΔT
γ.ΔT = 3α.ΔT
γ = 3α
β = 2α
A = A (1 + β.ΔT)
L2 = L2 (1 + α.ΔT)2
A = L2 (1 + 2α.ΔT + α2.ΔT2)
A = A (1 + 2α.ΔT)
A (1 + β.ΔT) = A (1+ 2α.ΔT)
Since α2.ΔT2 has a smaller volume, it is not considered
β = 2α
α : β : γ = 1 : 2 : 3
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Answer:
The relation between alpha, beta, and gamma is given in the form of a ratio and the ratio is 1:2:3 and can be expressed as:
alpha=fracbeta2=fracgamma3
Following is the relation between the three:
L = L (1 + α.ΔT)
Where, α is the coefficient of linear expansion
A = A (1 + β.ΔT)
Where, β is the coefficient of aerial expansion
V = V (1 + γ.ΔT)
Where, γ is the coefficient of cubical expansion
V = V + γV.ΔT
V = V (1 + γ.ΔT)
L3 = L3 (1 + α.ΔT)3
L3 = L3 (1 + 3α.ΔT + 3α2.ΔT2 + α3.ΔT3)
L3 = L3 (1 + 3α.ΔT)
Since 3α2.ΔT2 and α3.ΔT3 are smaller than 1, we are not considering them.
L3 = L3 (1 + 3α.ΔT)
V = L3 (1 + 3α.ΔT)
V (1 + γ.ΔT) = V (1 + 3α.ΔT)
1 + γ.ΔT = 1 + 3α.ΔT
γ.ΔT = 3α.ΔT
γ = 3α
β = 2α
A = A (1 + β.ΔT)
L2 = L2 (1 + α.ΔT)2
A = L2 (1 + 2α.ΔT + α2.ΔT2)
A = A (1 + 2α.ΔT)
A (1 + β.ΔT) = A (1+ 2α.ΔT)
Since α2.ΔT2 has a smaller volume, it is not considered
β = 2α
α : β : γ = 1 : 2 : 3