Understanding Thermal Expansion for a Steel Tire

This problem involves the concept of thermal expansion, specifically linear expansion. When materials are heated, their dimensions tend to increase due to the increased kinetic energy of their atoms and molecules. We are applying this principle to fit a steel tire onto a wooden wagon wheel.

The steel tire is initially smaller than the wagon wheel. By heating the steel tire, we cause it to expand until its inside diameter is large enough to fit over the outside diameter of the wagon wheel. Once it cools, it will contract, creating a tight fit.

Applying the Linear Expansion Formula

The change in length (or diameter, in this case) due to thermal expansion is given by the formula:

\(\Delta L = L_0 \alpha \Delta T\)

Where:

• \(\Delta L\) is the change in length (final length - initial length).
• \(L_0\) is the original length (initial diameter).
• \(\alpha\) is the coefficient of linear expansion of the material.
• \(\Delta T\) is the change in temperature (final temperature - initial temperature).

In this problem, we are given the following information:

• Original inside diameter of the steel tire, \(L_0 = 3737\) mm.
• Required final inside diameter of the steel tire to fit over the wagon wheel, \(L = 3750\) mm.
• Initial temperature of the steel tire, \(T_0 = 20^\circ\)C.
• Coefficient of linear expansion of steel, \(\alpha = 1.2 \times 10^{-5}/^\circ\)C.

Calculating the Required Diameter Change

First, let's find the required change in diameter (\(\Delta L\)) for the steel tire to fit over the wagon wheel:

\(\Delta L = L - L_0\)

\(\Delta L = 3750 \text{ mm} - 3737 \text{ mm}\)

\(\Delta L = 13 \text{ mm}\)

The steel tire needs to expand by 13 mm in diameter.

Determining the Temperature Change (\(\Delta T\))

Now, we can use the linear expansion formula to find the required change in temperature (\(\Delta T\)). We rearrange the formula to solve for \(\Delta T\):

\(\Delta T = \frac{\Delta L}{L_0 \alpha}\)

Substitute the known values into the formula:

\(\Delta T = \frac{13 \text{ mm}}{(3737 \text{ mm}) \times (1.2 \times 10^{-5}/^\circ\text{C})}\)

Let's calculate the denominator first:

\(3737 \times 1.2 \times 10^{-5} = 0.044844\)

So, the equation becomes:

\(\Delta T = \frac{13}{0.044844} \ ^\circ\text{C}\)

\(\Delta T \approx 289.82 \ ^\circ\text{C}\)

This is the required temperature change, or the increase in temperature from the initial temperature of 20°C.

Calculating the Final Temperature for Fitting the Tire

The question asks for the final temperature to which the steel tire must be heated. The final temperature \(T\) is the initial temperature \(T_0\) plus the temperature change \(\Delta T\):

\(T = T_0 + \Delta T\)

\(T = 20^\circ\text{C} + 289.82^\circ\text{C}\)

\(T \approx 309.82^\circ\text{C}\)

Rounding to the nearest degree or considering typical options in such problems, this value is very close to 310°C.

Therefore, the steel tire must be heated to approximately 310°C for it to fit over the wagon wheel.

Summary of Calculation

The temperature change needed is about 290°C, starting from 20°C, leading to a final temperature of about 310°C. This demonstrates the practical application of thermal expansion in fitting parts like a steel tire onto a wagon wheel.