Answer:
The length of the steel beam is 1.058 m
Explanation:
coefficient of linear expansivity of Aluminum,
= 25 x 10⁻⁶ ⁰C⁻¹
coefficient of linear expansivity of steel,
= 12 x 10⁻⁶ ⁰C⁻¹
Change in length of aluminum; ΔL
=
*
*ΔT
Change in length of steel; ΔL
=
*
*ΔT
difference in length of Aluminum and steel;
-
= 0.55 m, for this difference to remain constant, then ΔL
= ΔL
From the equation above,
= 0.55 +![L_{Al_i}](https://tex.z-dn.net/?f=L_%7BAl_i%7D)
Since, ΔL
= ΔL
, then
*
*ΔT =
*
*ΔT
At constant temperature, the equation becomes;
*
=
*
Recall; ![L_{st_i} = 0.55 + L_{Al_i}](https://tex.z-dn.net/?f=L_%7Bst_i%7D%20%3D%200.55%20%2B%20L_%7BAl_i%7D)
![(0.55 +L_{Al_i})\alpha_{st} = L_{Al_i}\alpha _{Al}\\\\0.55\alpha_{st} + L_{Al_i}\alpha_{st} = L_{Al_i}\alpha _{Al}\\\\ L_{Ali}\alpha _{Al} - L_{Al_i}\alpha_{st} = 0.55\alpha_{st}\\\\L_{Al_i} (\alpha _{Al} -\alpha_{st}) = 0.55\alpha_{st}\\\\L_{Al_i} =\frac{0.55\alpha_{st}}{\alpha _{st} -\alpha_{Al}} = \frac{0.55X12X10^{-6}}{(25X10^{-6}) -(12X10^{-6})}\\\\L_{Al_i} = \frac{0.55X12X10^{-6}}{13X10^{-6}} =0.508 m](https://tex.z-dn.net/?f=%280.55%20%2BL_%7BAl_i%7D%29%5Calpha_%7Bst%7D%20%3D%20L_%7BAl_i%7D%5Calpha%20_%7BAl%7D%5C%5C%5C%5C0.55%5Calpha_%7Bst%7D%20%2B%20L_%7BAl_i%7D%5Calpha_%7Bst%7D%20%3D%20L_%7BAl_i%7D%5Calpha%20_%7BAl%7D%5C%5C%5C%5C%20L_%7BAli%7D%5Calpha%20_%7BAl%7D%20-%20L_%7BAl_i%7D%5Calpha_%7Bst%7D%20%3D%200.55%5Calpha_%7Bst%7D%5C%5C%5C%5CL_%7BAl_i%7D%20%28%5Calpha%20_%7BAl%7D%20-%5Calpha_%7Bst%7D%29%20%3D%200.55%5Calpha_%7Bst%7D%5C%5C%5C%5CL_%7BAl_i%7D%20%3D%5Cfrac%7B0.55%5Calpha_%7Bst%7D%7D%7B%5Calpha%20_%7Bst%7D%20-%5Calpha_%7BAl%7D%7D%20%3D%20%5Cfrac%7B0.55X12X10%5E%7B-6%7D%7D%7B%2825X10%5E%7B-6%7D%29%20-%2812X10%5E%7B-6%7D%29%7D%5C%5C%5C%5CL_%7BAl_i%7D%20%3D%20%5Cfrac%7B0.55X12X10%5E%7B-6%7D%7D%7B13X10%5E%7B-6%7D%7D%20%3D0.508%20m)
To calculate the length of the steel beam;
![L_{st_i} = 0.55 + L_{Al_i}](https://tex.z-dn.net/?f=L_%7Bst_i%7D%20%3D%200.55%20%2B%20L_%7BAl_i%7D)
![L_{st_i} = 0.55 + 0.508 = 1.058 m](https://tex.z-dn.net/?f=L_%7Bst_i%7D%20%3D%200.55%20%2B%200.508%20%3D%201.058%20m)
Therefore, the length of the steel beam is 1.058 m