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Found in: Page 359

### Physics Principles with Applications

Book edition 7th
Author(s) Douglas C. Giancoli
Pages 978 pages
ISBN 978-0321625922

# (I) Super Invar™, an alloy of iron and nickel, is a strong material with a very low coefficient of thermal expansion $$\alpha = 0.20 \times 1{0^{ - 6}}\;/^\circ C$$. A 1.8-m-long tabletop made of this alloy is used for sensitive laser measurements where extremely high tolerances are required. How much will this alloy table expand along its length if the temperature increases 6.0 C°? Compare to tabletops made of steel.

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 is $$1.30 \times {10^{ - 4}}{\rm{ m}}$$. The change in the length of the steel table is approximately 60 times the change in the length of the alloy table

See the step by step solution

## Step 1: Identification of given data

• 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 $$\Delta {T_{st}} = 6^\circ {\rm{C}}$$.

## Step 2: Understanding the increase in the length of the alloy table

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.

## Step 3: Determination of the change in the length of the alloy table

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 $$2.16 \times {10^{ - 6}}{\rm{ m}}$$.

## Step 4: Determination of the change in the length of the steel table

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 $$1.30 \times {10^{ - 4}}{\rm{ m}}$$.

## Step 5: Determination of the ratio of the change in the length of the steel table to the change in the length of the alloy table

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.

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