∙(a) Steel train rails are laid in 12.0 -m-long segments...

$\bullet$ (a) Steel train rails are laid in 12.0 -m-long segments placed end to end. The rails are laid on a winter day when their temperature is $-2.00^{\circ} \mathrm{C}$ . How much space must be left between adjacent rails if they are just to touch on a summer day when their temperature is $33.0^{\circ} \mathrm{C} ?$ (b) If the rails are mistakenly laid in contact with each other, what is the stress in them on a summer day when their temperature is $33.0^{\circ} \mathrm{C}$ ?

Key Concepts

Linear Thermal Expansion

Linear thermal expansion is the tendency of a material to change its dimensions in response to a change in temperature. The change in length of an object due to a temperature variation is typically given by ?L = L???T, where L? is the original length, ? is the coefficient of linear expansion, and ?T is the change in temperature. This concept is crucial for understanding how materials such as steel rails expand in hot conditions and contract in cold conditions.

Coefficient of Linear Expansion

The coefficient of linear expansion (?) quantifies how much a material expands or contracts per unit length for a one-degree change in temperature. It is specific to the material and is usually expressed in units of 1/°C. This parameter is essential in calculating the length change in the rails when their temperature goes from a cold winter day to a hot summer day.

Thermal Stress

Thermal stress arises in a material when its thermal expansion or contraction is restricted, preventing it from expanding freely or contracting accordingly. When a material is forced to maintain its original dimensions in spite of a temperature change, internal forces develop. This stress can be computed using Hooke’s law, where stress is proportional to strain with Young's modulus as the constant of proportionality.

Young's Modulus

Young's modulus is a measure of the stiffness of a material and defines the relationship between stress (force per unit area) and strain (relative deformation) in the material. When thermal expansion is constrained, the induced thermal strain (which would otherwise be ??T) directly leads to a corresponding stress given by ? = E??T, where E is Young's modulus. This concept helps in determining the stress in the rails when they are laid in contact.

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