Dynamics classification: Which of the following is best modeled as a first-order system under standard assumptions?

Introduction / Context:
First-order and second-order dynamic models are fundamental in control engineering. Recognizing the correct order helps in selecting appropriate controllers, predicting response times, and understanding overshoot and damping. Classic textbook examples illustrate these categories clearly.

Given Data / Assumptions:

• Lumped parameters, linearized behavior around an operating point.
• Thermal resistance–capacitance (RC) analogy for thermometers.
• No significant transport delay or secondary dynamics for the thermometer case.

Concept / Approach:

A mercury-in-glass thermometer responds to a step change in ambient temperature according to a single energy balance: heat flux through the bulb wall (thermal resistance) changes the mercury temperature (thermal capacitance). This yields a first-order differential equation with time constant τ, producing an exponential approach to the new temperature without oscillation.

Step-by-Step Solution:

Verification / Alternative check:

Experimental step tests of thermometers show a single-exponential response characterized by τ (e.g., 63.2% response at one time constant), consistent with a first-order model.

Why Other Options Are Wrong:

Damped vibrator: Mass-spring-damper is second-order. Interacting two-tank system: Hydraulically coupled states yield a second-order model. Non-interacting two tanks: Still results in the product of two first-order elements (overall second-order), each tank contributes its own time constant.

Common Pitfalls:

Assuming “two first-order elements in series” must be first-order; overall it is second-order and may show sluggish or weakly oscillatory behavior if additional dynamics exist.

Final Answer:

Mercury-in-glass thermometer suddenly immersed in boiling water