Back in the stage coach era, iron rims were placed on wooden wheels by shrink-fitting. Suppose that the wheel has a diameter of 100.00 cm, whereas the rim has an inside diameter if 99.75 cm at 20°C. How hot must the iron rim be heated if it is to be slid over the wheel?
(Let's say the average coefficient of linear expansion for iron is 11 x 10^-6 [(° C)^-1] )
To determine how hot the iron rim must be heated, we need to consider the concept of linear expansion, specifically the equation:
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.
In this case, we can treat the rim as a cylindrical object, and the length mentioned in the equation can be related to the circumference of the rim, given by 2πr.
First, we need to find the initial length of the rim at 20°C:
Circumference = 2πr
Circumference = 2π(99.75/2) [since the inside diameter is given]
Next, we can calculate the change in length or expansion of the rim when heated:
ΔL = α * L * ΔT
= α * Circumference * ΔT
We know the value of α (coefficient of linear expansion) for iron is 11 x 10^-6 [(°C)^-1], and we need to find ΔT.
Rearranging the equation, we get:
ΔT = ΔL / (α * Circumference)
Substituting the values:
ΔT = 100.00 cm / (11 x 10^-6 [(°C)^-1] * Circumference)
Calculate Circumference using the formula above, and then substitute it into the equation to find ΔT.
Finally, ΔT will give us the required temperature increment to reach the desired inside diameter of the rim.