Mechanical thermostats for home heating, and flasher units for car turn-signals (and Christmas tree lights!) often use this principle: Two strips, copper and iron, are bonded together, side by side. They share the same elastic properties (e.g., Young?s modulus), and the same length l, and diameter d. They don't have the same linear thermal-expansion coefficient, though ? if we heat them both to a higher temperature, one expands more than the other, and they bend. If the change in temperature is delta T and the linear thermal-expansion coefficients (alpha Fe) and (alpha Cu) , find the angle of deflection.

To understand how mechanical thermostats and flasher units work using the principle of thermal expansion, we need to delve into the mechanics of two different metals—copper and iron—bonded together. When heated, these metals expand at different rates due to their distinct linear thermal expansion coefficients. This difference in expansion causes the bonded strip to bend, which is the fundamental principle behind these devices. Let's break down the process step by step to find the angle of deflection.

Understanding Thermal Expansion

Every material expands when heated, and the degree to which it expands is quantified by its linear thermal expansion coefficient (α). For copper and iron, these coefficients are different:

• α_Cu (Copper): Approximately 16.5 x 10^-6 /°C
• α_Fe (Iron): Approximately 11 x 10^-6 /°C

This means that for the same temperature change (ΔT), copper will expand more than iron.

Calculating the Difference in Expansion

When both metals are subjected to a temperature change ΔT, the change in length (ΔL) for each metal can be calculated using the formula:

ΔL = L₀ × α × ΔT

Where:

• L₀ = original length of the metal strip
• α = linear thermal expansion coefficient
• ΔT = change in temperature

For copper and iron, the changes in length will be:

• ΔL_Cu = L₀ × α_Cu × ΔT
• ΔL_Fe = L₀ × α_Fe × ΔT

Finding the Bending Angle

The difference in expansion causes the strip to bend. The curvature (κ) of the strip can be related to the difference in lengths:

ΔL_Cu - ΔL_Fe = (L₀ × (α_Cu - α_Fe) × ΔT)

Assuming the strip bends into an arc, we can relate the curvature to the angle of deflection (θ) using the radius of curvature (R):

κ = 1/R = (ΔL_Cu - ΔL_Fe)/L₀

Thus, we can express the angle of deflection as:

θ = (ΔL_Cu - ΔL_Fe) / (L₀ × R)

Final Expression for Angle of Deflection

Substituting the expressions for ΔL into the equation gives:

θ = (L₀ × (α_Cu - α_Fe) × ΔT) / (L₀ × R)

After simplifying, we find:

θ = (α_Cu - α_Fe) × ΔT / R

Practical Application

This principle is crucial in various applications, such as in thermostats where the bending of the metal strip can open or close electrical contacts, thereby regulating temperature. Similarly, in automotive turn signals, the bending action can activate the light, providing a visual indication of a turn.

In summary, the angle of deflection of the bonded copper and iron strip due to a temperature change can be calculated using their thermal expansion coefficients and the radius of curvature. This fascinating interplay of materials and thermal physics is what makes these everyday devices function effectively!