Two straight metallic stripes each of thickness t and length l are riveted together.Their coefficients of linear expansion are α₁ and α₂.If they are heated through a temperature Δt,the bimetallic stripe will bend to form an arc of radius

a. t(Δt)/(α₁+α₂)

b. t(Δt)/(α₂-α₁)

c. t(α₁+α₂)Δt

d. t(α₂-α₁)Δt

To understand how two metallic stripes riveted together behave when heated, we need to consider the concept of thermal expansion. When materials are heated, they expand, and if two materials with different coefficients of linear expansion are joined together, they will expand by different amounts. This difference in expansion causes the bimetallic strip to bend. Let's break down the problem step by step.

Understanding Thermal Expansion

Each material has a coefficient of linear expansion, denoted as α. This coefficient indicates how much a unit length of the material will expand per degree of temperature change. For our two metallic stripes, we have:

• Stripe 1: Coefficient of linear expansion = α 1
• Stripe 2: Coefficient of linear expansion = α 2

Behavior Upon Heating

When the temperature increases by Δt, the change in length for each stripe can be expressed as:

• Change in length of Stripe 1 = α 1 * l * Δt
• Change in length of Stripe 2 = α 2 * l * Δt

Since these two stripes are riveted together, they cannot expand freely. Instead, the difference in their expansions will cause the bimetallic strip to bend. The stripe with the higher coefficient of expansion will elongate more than the one with the lower coefficient.

Calculating the Radius of Curvature

The bending of the bimetallic strip can be modeled as forming an arc. The radius of curvature (a) of this arc can be derived from the difference in expansion of the two materials. The formula for the radius of curvature in terms of the thickness (t) of the strips, the temperature change (Δt), and the coefficients of linear expansion is given by:

Deriving the Formula

The difference in expansion causes a bending moment, and the radius of curvature can be expressed as:

• Radius of curvature, a = (t * Δt) / (difference in expansion per unit length)

The difference in expansion per unit length can be expressed as:

• Difference = α 2 - α 1

Substituting this into our formula gives:

• a = (t * Δt) / (α 2 - α 1)

Final Answer

Thus, the correct expression for the radius of curvature when the two metallic stripes are heated is:

a = t(Δt) / (α 2 - α 1)

This means the answer corresponds to option b. Understanding this principle is crucial in applications like thermostats and other devices that utilize bimetallic strips for temperature regulation.