the length of an aluminum is 76.5cm and that of an iron ruler is 80cm at 0¡ãc. at what temperature will the length of the rules become equal
To determine the temperature at which the lengths of the aluminum and iron rulers become equal, we can use the concept of thermal expansion.
The equation that relates the change in length (∆L) of a material to the original length (L0), coefficient of linear expansion (α), and change in temperature (∆T) is given by:
∆L = α * L0 * ∆T
In this case, we need to find the temperature (∆T) at which the lengths of the two rulers become equal (∆L_aluminum = ∆L_iron).
Let's assume α_aluminum and α_iron represent the coefficients of linear expansion for aluminum and iron, respectively.
For aluminum ruler:
∆L_aluminum = α_aluminum * L0_aluminum * ∆T_aluminum
For iron ruler:
∆L_iron = α_iron * L0_iron * ∆T_iron
Since the lengths become equal, we have:
∆L_aluminum = ∆L_iron
Substituting the values we know:
α_aluminum * L0_aluminum * ∆T_aluminum = α_iron * L0_iron * ∆T_iron
Now we can solve for ∆T_aluminum by rearranging the equation:
∆T_aluminum = (α_iron * L0_iron * ∆T_iron) / (α_aluminum * L0_aluminum)
Plugging in the values we have:
∆T_aluminum = (α_iron * 80 * ∆T_iron) / (α_aluminum * 76.5)
Given that we are looking for the temperature when the lengths are equal, ∆T_aluminum = ∆T_iron. Therefore, we can rewrite the equation as:
∆T = (α_iron * 80) / (α_aluminum * 76.5)
To find the temperature (∆T) at which the lengths of the rulers become equal, you will need the coefficients of linear expansion for aluminum and iron. These values can be typically found in reference materials or online databases. Once you have the coefficients, plug them into the equation above to calculate ∆T, which represents the temperature difference from 0¡ãC. Finally, add ∆T to 0¡ãC to find the temperature at which the lengths of the rulers become equal.