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13 lut 2018 · In volume expansion β ≈ 3α, let's suppose we have a rectangular solid with height ho, width wo, and length lo, then Vo = howolo. Now, after heating, each side increases by a factor of αΔT, V =ho(1 + αΔT)wo(1 + αΔT)lo(1 + αΔT) ΔV = V −Vo =howolo(1 + αΔT)3 −Vo. We will substitute first equation, since Vo =howolo.
23 lis 2009 · In summary, the conversation discusses the relationship between temperature, heat, and volume thermal expansion. The approximation \beta=3\alpha is explored and it is suggested to use definitions of linear and volumetric thermal expansion coefficients to solve the problem.
Definition: Thermal Expansion in Three Dimensions. The relationship between volume and temperature \(\frac{dV}{dT}\) is given by \(\frac{dV}{dT} = \beta V \Delta T\), where \(\beta\) is the coefficient of volume expansion. As you can show in Exercise, \(\beta = 3\alpha\). This equation is usually written as \[\Delta V = \beta V \Delta T.\]
Therefore, the coefficient of volume expansion is 3 times as much as the coefficient of the linear expansion. β= 3α. (20.3.3) (20.3.3) β = 3 α. 🔗. Bear in mind that this relation only works out when the volume change in small compared to the original volume.
(c) Prove that the volume thermal expansion coefficient of a solid is equal to the sum of its linear expansion coefficients in the three directions: β= α x + α
28 sie 2011 · Prove that the volume thermal expansion coefficient of a solid is equal to the sum of its linear expansion coefficients in the three directions. [itex]\beta[/itex]=[itex]\alpha[/itex]x +[itex]\alpha[/itex]y+[itex]\alpha[/itex]z
This relationship can be found by considering the thermal expansion of a simple body, such as a cube with an edge \(l_0\). When a cube heats up by \(\Delta{t}\), each side of the cube will increase by \(\Delta{l}\) and become equal to \( l ~= ~l_0(1 ~+ ~\alpha{\Delta{t}}) \) (7-3) The volume in this case equals
