Help with physics heat question!?

Question

Problem 14.10
A steel bar and a copper bar have the same length of 1.000 m at -20.00 ∘C.

Part A
What is the difference in the lengths of the two bars at 33.0 ∘C?
Express your answer in millimeters to three significant figures.

Answer 1

λs = 12.0*10^-6 °C^-1

λc = 17.0*10^-6 °C^-1

ΔL = 1000*((17-12)*10^-6*(33-(-20)) = 0.265 mm

Answer 2

A temperature change ΔT causes a change in any linear dimension L₀ of a solid body. The change ΔL is approximately proportional to L₀ and ΔT. The quantity α is the coefficient of linear expansion, and varies depending on the material you are using.



Let α𝒸 = coefficient of linear expansion of copper and let α𝓈 = coefficient of linear expansion of steel. Use a table to look up the coefficients (see link).



Copper bar:

ΔL = α𝒸L₀ΔT

ΔL = (16.6 × 10⁻⁶ ℃⁻¹)(1.000 m)[33.0℃ − (-20.00℃)]

ΔL = 8.798 × 10⁻⁴ m

ΔL = 8.80 × 10⁻⁴ m

ΔL = 0.880 mm



Steel bar:

ΔL = α𝓈L₀ΔT

ΔL = (12.0 × 10⁻⁶ ℃⁻¹)(1.000 m)[33.0℃ − (-20.00℃)]

ΔL = 6.36 × 10⁻⁴ m

ΔL = 0.636 mm



The copper bar is 0.880 mm − 0.636 mm = 0.244 mm longer than the steel bar. This makes sense because the coefficient of linear expansion for copper is greater than steel; therefore, we would expect the copper bar to be longer than the steel bar.



Solution: 0.244 mm

Answer 3

This is heavily dependent on what you take as the values for linear thermal expansion, and I have seen a range of values published for each.



I'll use

α_steel = 13e-6 /ºC

α_copper = 17e-6 /ºC



ΔL = α * L * ΔT, so

steel ΔL = 13e-6/ºC * 1.000m * 53.0º = 0.000689 m

copper ΔL = 17e-6/ºC * 1.000m * 53.0º = 0.000901 m

so the difference in lengths is 0.000212 m = 0.212 mm ◄



If for steel you use α = 12e-6ºC, the difference in lengths is 0.265 mm.



Hope this helps!