Answer:
To calculate two objects of different materials and different initial length to have the same length when heated, we can use the formula:
ΔL = αLΔT
where ΔL is the change in length, α is the coefficient of thermal expansion, L is the original length, and ΔT is the change in temperature.
Let's assume we have two objects: object 1 made of steel with an initial length of 1 meter and object 2 made of copper with an initial length of 0.5 meters. We want to find the temperature change required to make both objects have the same length.
First, we need to find the coefficient of thermal expansion for each material. The coefficient of thermal expansion for steel is approximately 1.2 x 10^-5 per degree Celsius, and the coefficient of thermal expansion for copper is approximately 1.7 x 10^-5 per degree Celsius.
Next, we can use the formula to solve for the temperature change required to make both objects have the same length:
ΔL1 = α1L1ΔT
ΔL2 = α2L2ΔT
We want ΔL1 + L1 = ΔL2 + L2, which means we want ΔL1 - ΔL2 = L2 - L1.
Substituting the equations above and solving for ΔT, we get:
ΔT = (L2 - L1) / (α1L1 - α2L2)
ΔT = (0.5 m - 1 m) / [(1.2 x 10^-5 per degree Celsius)(1 m) - (1.7 x 10^-5 per degree Celsius)(0.5 m)]
ΔT ≈ 178.57 degrees Celsius
Therefore, to make both objects have the same length when heated, we would need to increase their temperature by approximately 178.57 degrees Celsius