A 1.0-m thin rod of aluminum is laid end-to-end with a 1.0-m thin rod of steel. The rods
are initially at 25 oC. What is the total length of both rods when heated to 450 oC?
L=L₁+L₂=L₀₁+ΔL₁ +L₀₂ + ΔL ₂=
= L₀₁+α₁•L₀₁•ΔT +L₀₂ + α₂•L₀₂•ΔT=
=1+ 23.1•10⁻⁶(450-25) +1+ 12•10⁻⁶(450-25) =
=2.015 m
To find the total length of both rods when heated to 450 oC, we need to consider the thermal expansion of both aluminum and steel.
The formula for linear expansion is given by:
ΔL = α * L * ΔT
where:
ΔL = change in length,
α = coefficient of linear expansion,
L = initial length,
ΔT = change in temperature.
First, we need to calculate the change in length for each rod separately using the given values.
For aluminum:
Coefficient of linear expansion (α) for aluminum = 23 × 10^-6 / oC
Initial length (L) = 1.0 m
Change in temperature (ΔT) = 450 oC - 25 oC = 425 oC
ΔL_aluminum = α_aluminum * L * ΔT
ΔL_aluminum = 23 × 10^-6 / oC * 1.0 m * 425 oC
For steel:
Coefficient of linear expansion (α) for steel = 12 × 10^-6 / oC
Initial length (L) = 1.0 m
Change in temperature (ΔT) = 450 oC - 25 oC = 425 oC
ΔL_steel = α_steel * L * ΔT
ΔL_steel = 12 × 10^-6 / oC * 1.0 m * 425 oC
Now, we can find the change in length for each rod and add them to get the total change in length.
ΔL_total = ΔL_aluminum + ΔL_steel
Finally, we can find the total length of both rods when heated to 450 oC by adding the change in length to the initial length of the rods.
Total length = Initial length + ΔL_total
Please note that the specific values for the coefficients of linear expansion might differ slightly depending on the reference source.
To find the total length of both rods when heated to 450 °C, we need to take into account the thermal expansion of aluminum and steel.
The formula for linear thermal expansion is given by:
ΔL = α * L * ΔT,
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.
First, let's calculate the change in length for each rod separately using the formula. We need to know the coefficient of linear expansion for both aluminum and steel:
The coefficient of linear expansion for aluminum (α_aluminum) is 23.1 x 10^-6 /°C.
The coefficient of linear expansion for steel (α_steel) is 12.0 x 10^-6 /°C.
For the aluminum rod:
ΔL_aluminum = α_aluminum * L_aluminum * ΔT
= 23.1 x 10^-6 /°C * 1.0 m * (450 °C - 25 °C)
= 23.1 x 10^-6 /°C * 1.0 m * 425 °C
For the steel rod:
ΔL_steel = α_steel * L_steel * ΔT
= 12.0 x 10^-6 /°C * 1.0 m * (450 °C - 25 °C)
= 12.0 x 10^-6 /°C * 1.0 m * 425 °C
Next, we can find the total change in length for both rods combined:
ΔL_total = ΔL_aluminum + ΔL_steel
Finally, we can calculate the total length of both rods when heated to 450 °C:
Total length = L_aluminum + ΔL_total
By substituting the values for the coefficients of linear expansion, original lengths of the rods, and the change in temperature, we can solve for the total length.