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 -9.0°C. (a) 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°C?

Verified step by step guidance

1. Identify the relevant physical quantities: initial temperature (T_i = -9.0°C), final temperature (T_f = 33.0°C), and the length of each rail segment (L = 12.0 m).
2. Understand that the problem involves thermal expansion, which describes how the size of an object changes with a change in temperature. The formula for linear expansion is ΔL = αLΔT, where ΔL is the change in length, α is the coefficient of linear expansion for steel, L is the original length, and ΔT is the change in temperature.
3. Calculate the change in temperature, ΔT = T_f - T_i.
4. Look up the coefficient of linear expansion for steel, typically around α = 11 x 10^-6 /°C.
5. Substitute the values into the thermal expansion formula to find ΔL, the change in length of each rail segment. This will give the minimum space needed between the rails to just touch on a hot day.

Key Concepts

Thermal Expansion

Thermal expansion refers to the tendency of materials to change in size or volume in response to temperature changes. For solids, this is typically linear expansion, where the length of an object increases as its temperature rises. The amount of expansion can be calculated using the formula ΔL = αL₀ΔT, where ΔL is the change in length, α is the coefficient of linear expansion, L₀ is the original length, and ΔT is the change in temperature.

Coefficient of Linear Expansion

The coefficient of linear expansion (α) is a material-specific constant that quantifies how much a material expands per degree of temperature increase. For steel, this value is approximately 11 x 10⁻⁶ /°C. Understanding this coefficient is crucial for calculating the total expansion of the steel rails when the temperature changes from winter to summer conditions.

Temperature Difference

The temperature difference (ΔT) is the difference between two temperatures, which in this case is the summer temperature (33.0°C) minus the winter temperature (-9.0°C). This difference is essential for calculating the total expansion of the steel rails, as it directly influences how much space must be left between adjacent rails to accommodate the expansion without causing deformation or buckling.