A steel rod 4.00 cm in diameter is heated so that its...

A steel rod $4.00 \mathrm{~cm}$ in diameter is heated so that its temperature increases by $70.0^{\circ} \mathrm{C}$. It is then fastened between two rigid supports. The rod is allowed to cool to its original temperature. Assuming that Young's modulus for the steel is $20.6 \times 10^{10} \mathrm{~N} / \mathrm{m}^{2}$ and that its average coefficient of linear expansion is $11.0 \times 10^{-6}\left({ }^{\circ} \mathrm{C}\right)^{-1}$, calculate the tension in the rod.

Key Concepts

Stress-Strain Relationship

The stress-strain relationship in elasticity defines how much internal stress develops for a given strain produced by external factors, such as thermal expansion or contraction. When a material is constrained from freely expanding or contracting due to temperature changes, the resulting strain is directly proportional to the induced stress, with the constant of proportionality being Young’s modulus.

Cross-Sectional Area Calculation

The cross-sectional area is a critical factor in translating a material’s stress into an overall force or tension. For cylindrical objects, such as rods, the area is calculated using the formula for the area of a circle. This area is then used in conjunction with the calculated stress to determine the total force developed in the material when thermal deformation is constrained.

Thermal Expansion

Thermal expansion refers to the tendency of materials to change in size—typically to expand—when their temperature changes. This concept is quantified by the coefficient of linear expansion, which indicates the fractional change in length per degree change in temperature. In problems where an object is heated and then constrained as it cools, the prevented free contraction or expansion can lead to internal stresses.

Elasticity and Young's Modulus

Young's modulus is a material property that defines the relationship between stress (force per unit area) and strain (relative deformation) within the elastic limit of the material. It is used to calculate how much a material will deform under a given load and is essential in determining the stress generated when a material's natural thermal expansion is hindered.

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