A simple one
Thermo challenge
A copper and a tungsten plate having a thickness of 2 mm each are riveted together at 0⁰C 2 form a bimetallic strip
Find the average radius of curvature of this plate at 200⁰C
The coefficients of linear expansion for copper and tungsten are
1.7*10^-5 and.4*10^-5 K^-1 respectively

edited:10^-5K^-1 not 10^-1K^-1

To solve the problem of finding the average radius of curvature of a bimetallic strip made of copper and tungsten when heated from 0 °C to 200 °C, we can use the concept of thermal expansion and the formula for the curvature of a bimetallic strip. Let's break this down step by step.

Understanding the Components

First, we need to understand the materials involved:

• Copper: Coefficient of linear expansion, α_Cu = 1.7 × 10^-5 K^-1
• Tungsten: Coefficient of linear expansion, α_W = 0.4 × 10^-5 K^-1

Calculating the Change in Length

When the temperature increases, both materials will expand, but they will do so at different rates due to their different coefficients of linear expansion. The change in length (ΔL) for each material can be calculated using the formula:

ΔL = L₀ × α × ΔT

Where:

• L₀ is the original length (which we can assume to be 1 unit for simplicity),
• α is the coefficient of linear expansion,
• ΔT is the change in temperature (200 °C - 0 °C = 200 °C).

Calculating the Length Changes

For copper:

ΔL_Cu = 1 × (1.7 × 10^-5) × 200 = 3.4 × 10^-3 m

For tungsten:

ΔL_W = 1 × (0.4 × 10^-5) × 200 = 8.0 × 10^-4 m

Finding the Difference in Expansion

The difference in expansion between the two materials will cause the bimetallic strip to bend. The difference in length (ΔL_diff) is:

ΔL_diff = ΔL_Cu - ΔL_W = (3.4 × 10^-3) - (8.0 × 10^-4) = 2.6 × 10^-3 m

Calculating the Radius of Curvature

The radius of curvature (R) of the bimetallic strip can be approximated using the formula:

R = (t₁ + t₂) / (ΔL_diff / (t₁ * t₂))

Where:

• t₁ = thickness of copper = 2 mm = 0.002 m
• t₂ = thickness of tungsten = 2 mm = 0.002 m

Substituting the values:

R = (0.002 + 0.002) / ((2.6 × 10^-3) / (0.002 * 0.002))

R = 0.004 / (2.6 × 10^-3 / 4 × 10^-6)

R = 0.004 / (650) ≈ 6.15 × 10^-6 m

Final Result

Thus, the average radius of curvature of the bimetallic strip at 200 °C is approximately 6.15 mm. This curvature results from the differential expansion of the two metals, which is a fundamental principle behind the operation of bimetallic strips in various applications, such as thermostats and temperature sensors.