# A uniform steel rod has length l at 0 degrees Celcius. Now one of its end is kept in ice (0...

## Question:

A uniform steel rod has length l at 0{eq}^{\circ} {/eq} C. Now one of its end is kept in ice (0{eq}^{\circ} {/eq} C) and the other end is kept in steam (100{eq}^{\circ} {/eq} C).

If the coefficient of thermal expansion of the rod is a, how much is the thermal expansion of the rod at steady state?

## Linear Thermal Expansion:

It states that unit change in the length of the specimen when there is a unit change of the temperature. Different material has a different coefficient of thermal expansion. It depends on the temperature difference.

Given data

• The length of the rod is: {eq}l\;{\rm{m}} {/eq}
• The temperature at one end of the rod is: {eq}{T_{0^\circ C}} = 0^\circ {\rm{C}} {/eq}
• The temperature at the other end of the rod is: {eq}{T_{100^\circ {\rm{C}}}} = 100^\circ {\rm{C}} {/eq}
• The coefficient of the thermal expansion is: {eq}a\;^\circ {{\rm{C}}^{{\rm{ - 1}}}} {/eq}

The expression for linear thermal expansion of the rod at steady state is given as:

{eq}\Delta L = la \times \left( {{T_{100^\circ {\rm{C}}}} - {T_{{\rm{0^\circ C}}}}} \right) {/eq}

Substitute the value in the above expression.

{eq}\begin{align*} \Delta L &= la \times \left( {100 - 0} \right)\\ \Delta L &= 100la\;{\rm{m}} \end{align*} {/eq}

Thus, the thermal expansion of the rod is {eq}100la\;{\rm{m}} {/eq}. 