Problem Statement
A bimetallic strip consists of brass and steel riveted together. Which way does it bend when heated? ($\alpha_{brass} = 2.0 \times 10^{-5}$ °C$^{-1}$, $\alpha_{steel} = 1.2 \times 10^{-5}$ °C$^{-1}$)
Given Information
- All quantities, constants, and constraints stated in the problem above
- Physical constants used as needed (see Concepts section)
Physical Concepts & Formulas
This problem draws on fundamental physical principles. The key is to identify which conservation law or field equation governs the system, then apply it systematically. Dimensional analysis can always be used to verify that the final answer has the correct units. Working from first principles — rather than memorising formulas — builds deeper understanding and allows tackling novel problems.
- Identify the relevant physical law (Newton’s laws, conservation of energy/momentum, Maxwell’s equations, etc.)
- State the mathematical form of that law as it applies here
- Check dimensions at every step: both sides of an equation must have the same units
Step-by-Step Solution
Problem Statement
A bimetallic strip consists of brass and steel riveted together. Which way does it bend when heated? ($\alpha_{brass} = 2.0 \times 10^{-5}$ °C$^{-1}$, $\alpha_{steel} = 1.2 \times 10^{-5}$ °C$^{-1}$)
Given Information
- $\alpha_{brass} = 2.0 \times 10^{-5}$ °C$^{-1}$
- $\alpha_{steel} = 1.2 \times 10^{-5}$ °C$^{-1}$
- Both strips have the same initial length and are firmly bonded
Physical Concepts & Formulas
When two metals with different $\alpha$ values are bonded, heating causes one to expand more than the other. Since they are constrained together, the strip bends so that the metal with higher $\alpha$ is on the outer (longer) side of the curve.
Step-by-Step Solution
Step 1: Compare expansion coefficients.
$$\alpha_{brass} = 2.0 \times 10^{-5} > \alpha_{steel} = 1.2 \times 10^{-5}$$
Step 2: Determine which strip wants to be longer.
Brass tries to expand more for the same $\Delta T$. If free, brass would be longer than steel after heating.
Step 3: Determine bending direction.
Since they are riveted together, brass must be on the outside (convex side) to accommodate its greater length. The strip curves with brass on the outside, steel on the inside.
Worked Calculation
For $\Delta T = 1°C$, $L = 0.1$ m:
$$\Delta L_{brass} = 0.1 \times 2.0 \times 10^{-5} = 2.0 \times 10^{-6} \text{ m}$$
$$\Delta L_{steel} = 0.1 \times 1.2 \times 10^{-5} = 1.2 \times 10^{-6} \text{ m}$$
Brass wants to be longer → bends toward steel side.
Answer
The strip bends with brass on the convex (outer) side — it curves toward the steel.
Physical Interpretation
This is the operating principle of bimetallic thermostats. As temperature rises, the strip bends and breaks an electrical contact, cutting heating power. The greater the difference in $\alpha$ values, the more sensitive the thermostat. Cooling reverses the bend.
Worked Calculation
Substituting all given numerical values with their units into the derived formula:
$$\text{Numerical result} = \text{given expression substituted with values}$$
Answer
$$\boxed{\Delta L_{steel} = 0.1 \times 1.2 \times 10^{-5} = 1.2 \times 10^{-6} \text{ m}}$$
Physical Interpretation
The answer should be checked for dimensional consistency and physical reasonableness: is the magnitude in the expected range for this type of problem? Does the answer change in the correct direction when parameters are varied (e.g., increasing mass should increase momentum, increasing distance should decrease field strength)? These sanity checks are as important as the calculation itself.
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