91Ó°ÊÓ

• The coefficient of thermal expansion of the alloy is\({\alpha _{al}} = 0.20 \times {10^{ - 6}}\;{\rm{/^\circ C}}\).
• The coefficient of thermal expansion of the steel is\({\alpha _s} = 12 \times {10^{ - 6}}\;{\rm{/^\circ C}}\).
• The length of the table top is\(l = 1.8{\rm{ m}}\).
• The increase in the temperature of the alloy table\(\Delta {T_{al}} = 6^\circ {\rm{C}}\).
• The increase in the temperature of the steel table \(\Delta {T_{st}} = 6^\circ {\rm{C}}\).

The change in the length of the alloy depends on the length of the table, the coefficient of thermal expansion of the alloy, and the change in temperatures of the alloy.

With an increase in the temperature of the alloy, there is an increase in the length of the alloy.

The change in the length of the alloy table can be expressed as

\(\Delta {l_{al}} = {\alpha _{al}}l\Delta {T_{al}}\).

Substitute the values in the above equation.

\(\begin{aligned}{c}\Delta {l_{al}} &= 0.20 \times {10^{ - 6}}\;{\rm{/^\circ C}} \times 1.8{\rm{ m}} \times 6^\circ {\rm{C}}\\ &= 3.6 \times {10^{ - 7}}{\rm{ m/^\circ C}} \times 6^\circ {\rm{C}}\\ &= 2.16 \times {10^{ - 6}}{\rm{ m}}\end{aligned}\)

Thus, the change in the length of the alloy table is \(2.16 \times {10^{ - 6}}{\rm{ m}}\).

The change in the length of the steel table can be expressed as

\(\Delta {l_{st}} = {\alpha _{st}}l\Delta {T_{st}}\).

Substitute the values in the above equation.

\(\begin{aligned}{c}\Delta {l_{st}} &= 12 \times {10^{ - 6}}\;{\rm{/^\circ C}} \times 1.8{\rm{ m}} \times 6^\circ {\rm{C}}\\ &= 21.6 \times {10^{ - 6}}{\rm{ m/^\circ C}} \times 6^\circ {\rm{C}}\\ \approx 1.30 \times {10^{ - 4}}{\rm{ m}}\end{aligned}\)

Thus, the change in the length of the steel table is \(1.30 \times {10^{ - 4}}{\rm{ m}}\).

The ratio of the change in the steel table to the change in the alloy table can be expressed as

\(\begin{aligned}{c}\frac{{\Delta {l_{st}}}}{{\Delta {l_{al}}}} &= \frac{{1.30 \times {{10}^{ - 4}}{\rm{ m}}}}{{2.16 \times {{10}^{ - 6}}{\rm{ m}}}}\\ \approx 60.\end{aligned}\)

Thus, the change in the length of the steel table is approximately 60 times the change in the length of the alloy table.