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, αCu = 1.7 × 10-5 K-1
- Tungsten: Coefficient of linear expansion, αW = 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:
ΔLCu = 1 × (1.7 × 10-5) × 200 = 3.4 × 10-3 m
For tungsten:
ΔLW = 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 (ΔLdiff) is:
ΔLdiff = ΔLCu - ΔLW = (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) / (ΔLdiff / (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.
