Physics Beam Length

Question

A steel beam being used in the construction of a skyscraper has a length of 33.000 m when delivered on a cold day at a temperature of 14.000°F. What is the length of the beam when it is being installed later on a warm day when the temperature is 93.000°F? (Give your answer to at least five significant figures.) 

Answer 1

The change in length measurements of an object due to thermal expansion is related to temperature change by a coefficient of linear thermal expansion. It is the fractional change in length per degree of temperature change. Assuming negligible effect of pressure, we may write:

∝_T=(1/L)(dL/dT)

Where ∝_T = Linear Coefficient of Thermal Expansion

L = Length

T = Temperature

If linear-expansion coefficient does not change much over the change in temperature ΔT, and the fractional change in length is small ΔL/L≪1, the change in Length (ΔL) can be estimated as

ΔL = L∝_TΔT

For this problem

∝_T for iron = 11.8 x 10^-6/°K

L=33 m

ΔT = (93-14)°F=43.889°K (convert to °K and subtract)

So,

ΔL = 33(11.8 x 10^-6/°K)(43.889°K) = 0.01709 m

So, final Length is original length plus the change in length.

L = 33 M + 0.01709 m = 33.01709 m = 33.017 m rounded to 5 significant digits

Answer 2

Steps:

1. find difference in temperature (83F - 14 F) = 79 K
2. multiply by linear expansion coefficient for steel
3. multiple by answer in #2
4. add #3 to original length

let me know if any question