Question

A steel rod of length \(1.0000 \mathrm{~m}\) and cross-sectional area \(5.00 \cdot 10^{-4} \mathrm{~m}^{2}\) is placed snugly against two immobile end points. The rod is initially placed when the temperature is \(0.00^{\circ} \mathrm{C}\). Find the stress in the rod when the temperature rises to \(40.0^{\circ} \mathrm{C}\).

Short answer

Answer: The stress in the steel rod when the temperature rises to 40.0°C is 9.6 x 10^7 N/m².

Step-by-step solution

Determine the linear expansion of the rod

Due to the temperature change, the steel rod expands. To find the linear expansion, we can use the formula: \(\Delta L = L_i \alpha \Delta T,\) where \(\Delta L\) is the change in length, \(L_i\) is the initial length, \(\alpha\) is the coefficient of linear expansion, and \(\Delta T\) is the change in temperature. The problem states that the rod's initial length is \(1.0000 \mathrm{~m}\) and the temperature change is from \(0.00^{\circ} \mathrm{C}\) to \(40.0^{\circ} \mathrm{C}\), so \(\Delta T = 40.0^{\circ} \mathrm{C}\). The coefficient of linear expansion for steel is approximately \(12 \times 10^{-6} \mathrm{~K}^{-1}\). Now, we can plug the values into the formula: \(\Delta L = 1.0000\mathrm{~m} \times 12\times 10^{-6} \mathrm{~K}^{-1} \times 40.0 \mathrm{~K}.\)

Calculate the change in length

Using the data from Step 1, we can now calculate the change in length of the rod: \(\Delta L = 1.00\mathrm{~m} \times 12\times 10^{-6} \mathrm{~K}^{-1} \times 40.0 \mathrm{~K} = 0.00048 \mathrm{~m}.\)

Find the stress in the rod

When a material is restricted from expanding (due to the rod being placed snugly against two immobile end points), it is under stress. The stress formula is given by: \(F = \frac{Y\Delta L}{L_i},\) where \(F\) is the stress, \(Y\) is the Young's modulus of the material, and \(L_i\) is the initial length of the rod. The Young's modulus for steel is approximately \(2.0 \times 10^{11} \mathrm{~N\backslash m^{2}}\). Using the values from Step 2 and the Young's modulus for steel, we can now find the stress in the rod: \(F = \frac{(2.0 \times 10^{11}\mathrm{~N\backslash m^{2}})(0.00048\mathrm{~m})}{1.00\mathrm{~m}}.\)

Calculate the stress

Ending the previous equation, we obtain the stress value in the rod: \(F = (2.0 \times 10^{11} \mathrm{~N\backslash m^{2}})(0.00048\mathrm{~m}) = 9.6 \times 10^{7} \mathrm{~N\backslash m^{2}}.\) In conclusion, the stress in the steel rod when the temperature rises to \(40.0^{\circ} \mathrm{C}\) is \(9.6 \times 10^{7} \mathrm{~N\backslash m^{2}}\).

Key concepts

Linear Expansion

Imagine a steel rod as it begins to warm up; you can visually see that it gradually starts to stretch out. This effect is known as linear expansion, a phenomenon where the length of a material changes with temperature. When a substance like our steel rod is heated, its particles jiggle more vigorously, moving slightly farther apart and resulting in an elongation of the material.

Linear expansion is particularly crucial in construction and manufacturing, where materials are often exposed to a range of temperatures. For example, in bridges or railways, gaps are left to account for this expansion and prevent structural damage.

Young's Modulus

Young's modulus is all about understanding how a material behaves when it's pushed or pulled. Also known as the modulus of elasticity, it quantifies the stiffness of a solid material. It’s the ratio of stress (force applied per unit area) to strain (deformation in the shape), and it tells us how much a material will stretch or compress under certain loads.

You can think of it as a measure of how stubborn a material is; the higher the Young's modulus, the less it likes to change shape. For engineers, knowing the Young's modulus of the materials they use is essential to design structures that can withstand various forces without deforming too much.

Coefficient of Linear Expansion

Each material is unique in how it responds to temperature changes, and this is where the coefficient of linear expansion comes into play. It's a number that tells us the rate at which a material expands per degree increase in temperature. In simple terms, the coefficient of linear expansion is like the expansion personality of a material.

When we calculate the change in length of a steel rod, for instance, we multiply the original length by this coefficient and the temperature change to get the total expansion. For students tackling thermal stress problems, getting this coefficient correct is key to finding accurate predictions for how much an object will expand or contract.

Temperature Change

Temperature change is the driving force behind the thermal expansion of materials. When the temperature goes up, the material gets hotter and expands; when it drops, the material cools and contracts. It’s important to note that this isn't about the actual temperature but the change in temperature – how much hotter or colder something gets.

Temperature change is measured in degrees – Celsius (°C), Fahrenheit (°F), or Kelvin (K). It affects everything from the pressure in your car’s tires to the functioning of a metal bridge. Understanding how materials respond to temperature change is essential for solving real-world engineering challenges and is a fundamental step in predicting thermal stress in structures.