Question

A blacksmith fixes an iron ring on the rim of the wooden wheel of a horse cart. The diameter of the rim and the iron ring are $5.243m\;$and $5.231m$, respectively at ${27^0}C$. To what temperature should the ring be heated so as to fit the rim of the wheel?
A. ${356^0}C$
B. ${218^0}C$
C. ${562^0}C$
D. ${259^0}C$

Answer 1

Thermal expansion happens when an object or body expands in the reaction to being heated. This phenomenon can usually occur in gases and liquids but in some special cases, it can also occur on the solids. A solid’s thermal properties are a very important aspect in the design of factories and products.

Complete step by step solution:
The amount of how much a material can expand can be explained by considering the fractional growth of the material per degree change in temperature. This coefficient is known as the coefficient of thermal expansion. This coefficient is used to predict the growth of materials in reaction to their heating or temperature change. The larger this coefficient the more the material will expand per degree temperature. The formula for the coefficient of the thermal expansion is given as,
$\alpha = \dfrac{{\Delta L}}{{{L_0} \times \Delta T}}$
Here, $\alpha$ is the linear coefficient of temperature.
$\Delta L$ is the change in the length of test specimen due to heating or cooling
${L_0}$ is the actual length of the specimen at room temperature.
$\Delta T$ is temperature change, °C
From the formula, we can rearrange the formula to get the temperature change as,
${T_2} = \dfrac{{{d_w} - {d_i}}}{{{d_i}\alpha }} + {T_1}$
Here ${d_w}$and ${d_i}$are the diameter of the rim and the iron ring, for which the values are $5.243m\;$and $5.231m$
We know $\alpha = 1.2 \times {10^{ - 5}}{K^{ - 1}}$for iron
Also, the initial temperature is given as ${T_1} = {27^0}C$
Substituting all the know values,
${T_2} = \dfrac{{5.243m\; - 5.231m}}{{5.231m \times 1.2 \times {{10}^{ - 5}}{K^{ - 1}}}} \times {27^0}C$
This gives, ${T_2} = {218^0}C$

Therefore the temperature needed for the ring to be heated to fit the rim of the wheel is ${218^0}C$. The correct option is B.

Note:
The linear thermal coefficient of expansion has many applications in the industry. They are used for design purposes. They help predict the dimensional behavior of all structures subject to temperature changes. It can also determine the thermal stresses which can occur and this can cause the failure of a solid area composed of different materials when it is objected to a temperature excursion.