Question

A 18.0-m-long bar of steel expands due to a temperature increase. A 10.0-m-long bar of copper also gets longer due to the same temperature rise. The two bars were originally separated by a gap of 1.1 cm. Assume the steel and copper bars are fixed on the ends.

α(Steel) = 13 x 10^-6 K^-1
α(Copper) = 16.5 x 10^-6 K^-1

1) Calculate the change in temperature if the gap is exactly "closed" by the expanding bars. (Express your answer to two significant figures.)


2) Calculate the distances that the steel stretches. (Express your answer to two significant figures.)


3) Calculate the distances that the copper stretches. (Express your answer to two significant figures.)

Answer 1

1) We can use the equation for linear thermal expansion:

ΔL = αLΔT

where ΔL is the change in length, α is the coefficient of linear thermal expansion, L is the original length, and ΔT is the change in temperature.

Let's assume that both bars expand by the same amount, so that the initial gap of 1.1 cm is exactly closed. We can set up an equation:

ΔL(steel) + ΔL(copper) = 1.1 cm

Using the equation above and the given coefficients of linear thermal expansion and original lengths, we can write:

α(steel)L(steel)ΔT + α(copper)L(copper)ΔT = 1.1 cm

Solving for ΔT, we get:

ΔT = 1.1 cm / (α(steel)L(steel) + α(copper)L(copper))
ΔT = 1.1 cm / (13 x 10^-6 K^-1 × 18.0 m + 16.5 x 10^-6 K^-1 × 10.0 m)
ΔT ≈ 139.9 K

Therefore, the change in temperature is approximately 139.9 K.

2) We can now use the equation for linear thermal expansion again to calculate the distances that the steel stretches:

ΔL(steel) = α(steel)L(steel)ΔT
ΔL(steel) = 13 x 10^-6 K^-1 × 18.0 m × 139.9 K
ΔL(steel) ≈ 0.0408 m

Therefore, the steel stretches by approximately 0.0408 m.

3) Similarly, we can calculate the distances that the copper stretches:

ΔL(copper) = α(copper)L(copper)ΔT
ΔL(copper) = 16.5 x 10^-6 K^-1 × 10.0 m × 139.9 K
ΔL(copper) ≈ 0.0230 m

Therefore, the copper stretches by approximately 0.0230 m.