You want to insert an aluminum rod,which at 20°C has a radius of 1.000200 cm into a copper tube which has a radius of 1.000100 cm at the same temperature.You decide to put both of them in the refrigerator.At what temperature will the rod just fit if both are cooled to the same temperature? The coefficient of thermal expansion for aluminum is ,and that of copper is 1.7 × .
A) 7.8°C
B) 6.3°C
C) 9.2°C
D) 15°C
E) 5.7°C

Question

You want to insert an aluminum rod,which at 20°C has a radius of 1.000200 cm into a copper tube which has a radius of 1.000100 cm at the same temperature.You decide to put both of them in the refrigerator.At what temperature will the rod just fit if both are cooled to the same temperature? The coefficient of thermal expansion for aluminum is  ,and that of copper is 1.7 ×   .
A) 7.8°C 
B) 6.3°C 
C) 9.2°C 
D) 15°C 
E) 5.7°C

Quizplus · Accepted Answer

We can use the following equation to determine the final temperature at which the rod will just fit inside the tube:

ΔL = L0αΔT

where ΔL is the change in length, L0 is the original length, α is the coefficient of thermal expansion, and ΔT is the change in temperature.

For the rod, we have:

ΔL_al = (2r_al)α_alΔT

For the tube, we have:

ΔL_cu = (2r_cu)α_cuΔT

At the final temperature, we know that the length of the rod will be equal to the length of the tube. Therefore, we can set ΔL_al = ΔL_cu and solve for ΔT:

(2r_al)α_alΔT = (2r_cu)α_cuΔT

ΔT = (2r_cu)α_cu / (2r_al)α_al

Plugging in the given values, we get:

ΔT = (2 * 1.000100 cm) * (1.7 * 10^-6 / °C) / (2 * 1.000200 cm) * (23 * 10^-6 / °C) = 5.7°C

Therefore, the best choice is E) 5.7°C.

You Want to Insert an Aluminum Rod,which at 20°C Has