At temperature T0, two metal strips of length l0 and...

At temperature $T_{0}$, two metal strips of length $l_{0}$ and thickness $d$, is bolted so that their ends coincide. The upper strip is made up of metal $A$ and have coefficient of expansion $\alpha_{A}$ and lower strip is made up of metal $B$ with coefficient of expansion $\alpha_{B} \cdot\left(\alpha_{A}>\alpha_{B}\right)$. When temperature of their blastic strip is in-seased from $T_{0}$ to $\left(T_{0}+\Delta T\right)$, on strip become longer than the other and blastic strip is bend in the form of a circle as shown in fig. Calculate the radius of furvature $R$ of the strip: (a) $R:=\frac{\left[2+\left(\alpha_{A}+\alpha_{B}\right) \Delta T\right] d}{2\left(\alpha_{A}-\alpha_{B}\right) \Delta T}$ (b) $R=\frac{\left[2-\left(\alpha_{A}+\alpha_{B}\right) \Delta T\right] d}{2\left(\alpha_{A}-\alpha_{B}\right) \Delta T}$ (c) $R=\frac{\left[2+\left(\alpha_{A}-\alpha_{B}\right) \Delta T\right] d}{2\left(\alpha_{A}-\alpha_{B}\right) \Delta T_{i}}$ (d) $R=\frac{\left[2-\left(\alpha_{A}-\alpha_{B}\right) \Delta T\right] d}{2\left(\alpha_{A}-\alpha_{B}\right) \Delta \mathcal{k}}$

At temperature $T_{0}$, two metal strips of length $l_{0}$ and thickness $d$, is bolted so that their ends coincide. The upper strip is made up of metal $A$ and have coefficient of expansion $\alpha_{A}$ and lower strip is made up of metal $B$ with coefficient of expansion $\alpha_{B} \cdot\left(\alpha_{A}>\alpha_{B}\right)$. When temperature of their blastic strip is in-seased from $T_{0}$ to $\left(T_{0}+\Delta T\right)$, on strip become longer than the other and blastic strip is bend in the form of a circle as shown in fig. Calculate the radius of furvature $R$ of the strip:
(a) $R:=\frac{\left[2+\left(\alpha_{A}+\alpha_{B}\right) \Delta T\right] d}{2\left(\alpha_{A}-\alpha_{B}\right) \Delta T}$
(b) $R=\frac{\left[2-\left(\alpha_{A}+\alpha_{B}\right) \Delta T\right] d}{2\left(\alpha_{A}-\alpha_{B}\right) \Delta T}$
(c) $R=\frac{\left[2+\left(\alpha_{A}-\alpha_{B}\right) \Delta T\right] d}{2\left(\alpha_{A}-\alpha_{B}\right) \Delta T_{i}}$
(d) $R=\frac{\left[2-\left(\alpha_{A}-\alpha_{B}\right) \Delta T\right] d}{2\left(\alpha_{A}-\alpha_{B}\right) \Delta \mathcal{k}}$

Key Concepts

Thermal Expansion

Thermal expansion refers to the tendency of materials to change their dimensions in response to changes in temperature. It is quantified by the coefficient of expansion, which represents the fractional change in length per unit change in temperature. This concept is fundamental in understanding how heating or cooling can induce stresses or deformations in materials.

Bimetallic Strip

A bimetallic strip is composed of two different metals bonded together, each having a distinct coefficient of thermal expansion. When the strip is subjected to a temperature change, the difference in expansion between the two metals causes the strip to bend. This principle is widely used in thermostats and temperature sensing devices.

Differential Expansion

Differential expansion occurs when two materials expand at different rates under the same thermal conditions. In the context of a bimetallic strip, this difference in expansion leads to a curvature as one layer elongates more than the other. Understanding this differential is key to predicting the bending behavior of the strip.

Radius of Curvature

The radius of curvature is a measure of how sharply a curved object bends. In the case of a bimetallic strip, it represents the radius of the circular arc that the strip approximates when it is heated. The determination of this radius involves understanding the relationship between thermal expansion, material properties, and geometry.

Mechanics of Materials

Mechanics of materials involves analyzing the behavior of solid objects under various loads and temperature changes. It provides the frameworks and equations necessary to relate the physical properties of materials, such as thickness and expansion coefficients, to their resulting deformations, such as the curvature observed in bimetallic strips.

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