# A blacksmith fixes iron ring on the rim of the wooden wheel of a horse cart. The diameter of the...

## Question:

A blacksmith fixes iron ring on the rim of the wooden wheel of a horse cart. The diameter of the rim and the iron ring are 6.243 m and 6.231 m, respectively at 27{eq}^{\circ} {/eq}C. To what temperature should the ring be heated so as to fit the rim of the wheel?

## Linear expansion:

The increase in the length of a material with the increase in temperature of a material is defined as linear expansion, and the proportion of the fractional change of the length to the change in temperature is known as the coefficient of linear expansion. Copper has the highest coefficient of linear expansion among all the elements.

Given Data:

• The initial length is, {eq}{L_1} = 6.231\;{\rm{m}} {/eq}
• The final length is, {eq}{L_2} = 6.243\;{\rm{m}} {/eq}
• The initial temperature is, {eq}{T_1} = 27^\circ {\rm{C}} {/eq}

The expression for the coefficient of line equation is,

{eq}{L_2} = {L_1} + {L_1}\alpha \Delta t {/eq}............(1)

Here, {eq}\alpha {/eq} is the coefficient of linear expansion of iron {eq}\left( {1.20 \times {{10}^{ - 5}}\;{{\rm{K}}^{ - 1}}} \right) {/eq}

Substitute the values in equation (1)

{eq}\begin{align*} 6.243 &= 6.231 + 6.231 \times 1.20 \times {10^{ - 5}} \times \left( {{T_2} - 27} \right)\\ {T_2} &= 187.48^\circ {\rm{C}} \end{align*} {/eq}

Thus, the temperature at which ring should be heated is {eq}187.48^\circ {\rm{C}} {/eq}.