For the different length steel beams, which will require a greater temperature increase to expand by 1.0 cm from its original length: the shorter beam, or the longer beam?
Let's rewrite the linear expansion equation in terms of L and ΔT, as these are different for the short and long steel beams, and have the thermal expansion coefficient α and expansion ΔL on the other side of the equation, as these quantities are both the same for both short and long beams:
L⋅ΔT = (∆L/α).
So the right-hand side of this equation is the same for both the short and long beams:
Lshort⋅ΔTshort = (∆L/α),
Llong⋅ΔTlong = (∆L/α),
so we can set the left-hand sides of both these equations equal to each other:
Lshort⋅ΔTshort = Llong⋅ΔTlong.
Since Lshort < Llong, then for the equality to hold, ∆Tshort > ∆Tlong, and thus the shorter beam will require a greater increase in temperature to expand the same amount as the longer beam.
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