The temperature at which the aluminum and brass rods just come into contact can be found using the concept of thermal expansion. By setting up an equation based on the change in length of each rod at the unknown temperature, we can solve for the temperature. The rods will just touch when the temperature decreases by approximately 0.43 °C from 0.0 °C.
To determine the temperature at which the aluminum and brass rods just come into contact, we can use the concept of thermal expansion. The rods will start to touch when their total lengths add up to the original separation distance between them. The change in length for each rod can be calculated using the formula:
ΔL = αLΔT
where ΔL is the change in length, α is the coefficient of linear expansion, L is the original length, and ΔT is the change in temperature. Since the base to which the rods are clamped undergoes negligible thermal expansion, we can assume that the original separation distance (0.024 cm) remains constant. Setting up an equation to solve for the unknown temperature:
Laluminum + Lbrass = 0.024 cm
(L0, aluminum + αaluminumL0, aluminumΔT) + (L0, brass + αbrassL0, brassΔT) = 0.024 cm
Substituting the known values:
(50.0 cm + αaluminum(50.0 cm)(ΔT)) + (50.0 cm + αbrass(50.0 cm)(ΔT)) = 0.024 cm
Simplifying and rearranging:
(αaluminum + αbrass)50.0 cm ΔT + 100.0 cm = 0.024 cm
ΔT = -100.0 cm / [(αaluminum + αbrass)50.0 cm]
Substituting the known values for αaluminum and αbrass:
ΔT = -100.0 cm / [(23.2 × 10-6 °C-1 + 19.0 × 10-6 °C-1)50.0 cm]
Calculating ΔT:
ΔT ≈ -0.43 °C
The negative sign indicates that the temperature needs to be decreased by 0.43 °C for the rods to come into contact. Therefore, the rods will just touch when the temperature decreases by approximately 0.43 °C from 0.0 °C.