Question

A steel wire is rigidly fixed along diameter of aluminium ring of radius \(\mathrm{R}\) as shown. Linear expansion coefficient of steel is half of linear expansion coefficient for aluminium. If temperature of system is increased by \(\Delta \theta\) then the thermal stress developed in steel wire is: \(\left(\alpha_{\mathrm{Al}}\right.\) is linear expansion coefficient for aluminium and Young's modulus for steel is \(Y\) ) (1) more than \(\mathrm{R} \alpha_{\mathrm{Al}} \Delta \theta \mathrm{Y}\) (2) less then R \(\alpha_{\mathrm{Al}} \Delta \theta \mathrm{Y}\) (3) Equal to \(R \alpha_{\mathrm{Al}} \Delta \theta Y\) (4) equal to \(2 R \alpha_{A 1} \Delta \theta Y\) (5) equal to \(4 \mathrm{R} \alpha_{\mathrm{Al}} \Delta \theta \mathrm{Y}\)

Short answer

Less than \(\text{R} \times \text{α}_{\text{Al}} \times \text{Δθ} \times \text{Y}\).

Step-by-step solution

- Understand the Problem

Given a steel wire fixed along the diameter of an aluminium ring with a radius of \(\text{R}\), calculate the thermal stress in the steel wire upon increasing the temperature by \(\text{Δθ}\). \(\text{α}_{\text{Al}}\) is the linear expansion coefficient of aluminium, and Young's modulus for steel is \(\text{Y}\). The linear expansion coefficient of steel is half of that for aluminium.

- Linear Expansion of Aluminium Ring

Upon increasing the temperature by \(\text{Δθ}\), the radius of the aluminium ring increases. The change in radius of the aluminium ring is given by: \[\text{ΔR}_{\text{Al}} = \text{R} \times \text{α}_{\text{Al}} \times \text{Δθ}\]

- Linear Expansion of Steel Wire

Similarly, for the steel wire, the change in length is: \[\text{ΔL}_{\text{Steel}} = \text{R} \times \frac{\text{α}_{\text{Al}}}{2} \times \text{Δθ}\]

- Difference in Expansion

The difference in expansion between the aluminium ring and steel wire is: \[\text{Δ}_{\text{diff}} = \text{ΔR}_{\text{Al}} - \text{ΔL}_{\text{Steel}} = \text{R} \times \text{α}_{\text{Al}} \times \text{Δθ} - \text{R} \times \frac{\text{α}_{\text{Al}}}{2} \times \text{Δθ} = \frac{\text{R} \times \text{α}_{\text{Al}} \times \text{Δθ}}{2}\]

- Thermal Stress Calculation

The steel wire is under thermal stress due to the restricted expansion. Therefore, the thermal stress in the steel wire is given by: \[\text{Stress}_{\text{Steel}} = \text{Y} \times \frac{\text{Δ}_{\text{diff}}}{\text{R}} = \text{Y} \times \frac{\frac{\text{R} \times \text{α}_{\text{Al}} \times \text{Δθ}}{2}}{\text{R}} = \frac{\text{Y} \times \text{α}_{\text{Al}} \times \text{Δθ}}{2}\]

- Comparing with Given Options

The calculated thermal stress in the steel wire is \(\frac{\text{Y} \times \text{α}_{\text{Al}} \times \text{Δθ}}{2}\). When comparing with the given options, the correct choice is: less than \(\text{R} \times \text{α}_{\text{Al}} \times \text{Δθ} \times \text{Y}\).

Key concepts

Thermal Expansion

Thermal expansion occurs when materials change their dimensions with temperature. When you heat an object, it expands; when you cool it, it contracts. This change in size is due to the increase in the kinetic energy of the particles within the material. They move more violently and take up more space.
The change in length is given by the formula: \[ \text{ΔL} = \text{L}_0 \times \text{α} \times \text{ΔT} \]Here - \text{ΔL\} = Change in length,- \text{L}_0\ = Original length,- \text{α\} = Linear expansion coefficient,- \text{ΔT\} = Change in temperature.
The kind of expansion discussed here is linear expansion, which specifically deals with the change along a single dimension like length or radius.
This principle is crucial in our problem with a steel wire and an aluminium ring. The linear expansion coefficients indicate how much each material will expand as the temperature increases.

Young's Modulus

Young's modulus is a measure of the stiffness of a material. It is a mechanical property that defines the relationship between stress (force per unit area) and strain (proportional deformation) in a material in the linear elasticity region of the material.
The formula for Young's modulus is:\[ \text{E} = \frac{\text{Stress}}{\text{Strain}} \]Stress is the force applied per unit area, and strain is the deformation experienced by the material compared to its original length.
In context, for the steel wire: - \text{Stress\} = Thermal stress due to temperature change,- \text{Strain\} = Relative change in length of steel compared to its original length because of restricted thermal expansion.
Knowing the Young's modulus (E) helps us calculate the material's response to thermal stress. The stiffer the material (higher E), the less it deforms when subjected to stress.

Linear Expansion Coefficient

The linear expansion coefficient, denoted as \text{α\}, is a material-specific value that quantifies how much a material expands per degree of temperature change.
For instance: if \text{α\} for aluminium is higher than that of steel, aluminium will expand more for the same temperature increase.
In the problem: The given data mentions that the linear expansion coefficient of steel is half of that for aluminium. Thus, while the aluminium ring's radius increases more significantly upon heating, the steel wire (fixed to the diameter) attempts to expand less, causing a discrepancy that results in thermal stress in the steel wire.
In mathematical terms, this is expressed as:\[ \text{ΔL}_{\text{material}} = \text{L}_0 \times \text{α}_{\text{material}} \times \text{ΔT} \]where each material expands differently depending on its respective linear expansion coefficient.
This variance in thermal expansion leads to the stress calculation provided in the problem's solution.