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Quiz: Physical System Modeling

Test your understanding of deriving mathematical models for electrical, mechanical, and electromechanical systems.

1. In an RC circuit with input voltage across both components and output taken across the capacitor, the transfer function is:

  1. $G(s) = \frac{RCs}{RCs + 1}$
  2. $G(s) = \frac{1}{RCs + 1}$
  3. $G(s) = \frac{RCs + 1}{1}$
  4. $G(s) = \frac{s}{RCs + 1}$
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The correct answer is B. The RC low-pass filter has transfer function $G(s) = 1/(RCs + 1) = 1/(\tau s + 1)$ where $\tau = RC$ is the time constant. This is a first-order system with DC gain of 1 and a single pole at $s = -1/RC$.

Concept Tested: RC Circuit

2. In the impedance analogy for analogous systems, force in a mechanical system corresponds to what quantity in an electrical system?

  1. Current
  2. Voltage
  3. Resistance
  4. Capacitance
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The correct answer is B. In the force-voltage (impedance) analogy, mechanical force corresponds to electrical voltage, velocity to current, and mass to inductance. This analogy allows engineers to apply electrical circuit analysis techniques to mechanical systems.

Concept Tested: Force-Voltage Analogy, Impedance Analogy

3. The transfer function of an armature-controlled DC motor (voltage input to angular velocity output) is typically:

  1. A pure integrator ($1/s$)
  2. A first-order system
  3. A second-order system
  4. A constant gain
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The correct answer is C. An armature-controlled DC motor with significant inductance has a second-order transfer function from voltage to velocity, with poles determined by electrical (armature) and mechanical (rotor inertia) time constants. If inductance is negligible, it simplifies to first-order.

Concept Tested: DC Motor, Motor Transfer Function

4. In a mass-spring-damper system, which component stores potential energy?

  1. The mass
  2. The spring
  3. The damper
  4. All three equally
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The correct answer is B. The spring stores potential energy ($E = \frac{1}{2}kx^2$). The mass stores kinetic energy ($E = \frac{1}{2}mv^2$). The damper dissipates energy (converts it to heat) but doesn't store it. Together, mass and spring give the system second-order dynamics.

Concept Tested: Mass-Spring-Damper

5. A gear train with gear ratio $N = N_2/N_1$ (output teeth / input teeth) transforms torques according to:

  1. $T_{out} = N \cdot T_{in}$
  2. $T_{out} = T_{in} / N$
  3. $T_{out} = T_{in}$
  4. $T_{out} = N^2 \cdot T_{in}$
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The correct answer is A. An ideal gear train multiplies torque by the gear ratio: $T_{out} = N \cdot T_{in}$. Conversely, angular velocity is divided by the gear ratio: $\omega_{out} = \omega_{in}/N$. Power (the product) is conserved in an ideal gear train.

Concept Tested: Gear Train

6. An RLC series circuit is analogous to which mechanical system?

  1. A pendulum
  2. A mass-spring-damper
  3. A lever system
  4. A fluid tank
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The correct answer is B. An RLC circuit and a mass-spring-damper share the same second-order differential equation structure. Using the force-voltage analogy: inductance L ↔ mass m, resistance R ↔ damping b, and 1/C ↔ spring constant k.

Concept Tested: RLC Circuit, Analogous Systems

7. In an op-amp inverting amplifier configuration, the output voltage is:

  1. Equal to the input voltage
  2. Proportional to the input with opposite sign
  3. The integral of the input
  4. The derivative of the input
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The correct answer is B. An inverting op-amp amplifier produces $V_{out} = -(R_f/R_{in}) \cdot V_{in}$, where $R_f$ is the feedback resistor and $R_{in}$ is the input resistor. The output is an inverted (negative) and scaled version of the input.

Concept Tested: Op-Amp Circuits

8. Thermal systems are typically modeled as:

  1. First-order systems with temperature as the state
  2. Second-order oscillatory systems
  3. Systems with complex conjugate poles
  4. Pure gain systems with no dynamics
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The correct answer is A. Thermal systems (like heating a room or a component) are typically first-order, with thermal resistance and capacitance creating a time constant $\tau = R_{th}C_{th}$. Temperature changes exponentially toward the new equilibrium, similar to an RC circuit.

Concept Tested: Thermal Systems

9. The mobility analogy differs from the impedance analogy in that:

  1. Force corresponds to current rather than voltage
  2. Only applies to rotational systems
  3. Ignores damping elements
  4. Cannot model capacitors
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The correct answer is A. In the mobility (force-current) analogy, force corresponds to current and velocity to voltage—the opposite of the impedance analogy. This alternative mapping can sometimes make circuit diagrams more intuitive for certain mechanical configurations.

Concept Tested: Mobility Analogy, Force-Current Analogy

10. A simple pendulum's equation of motion includes $\sin\theta$, which makes it:

  1. A linear system
  2. A first-order system
  3. A nonlinear system
  4. An unstable system
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The correct answer is C. The $\sin\theta$ term makes the pendulum equation nonlinear because $\sin\theta$ doesn't scale linearly with $\theta$. For small angles, we approximate $\sin\theta \approx \theta$ to obtain a linear model, but the true system is fundamentally nonlinear.

Concept Tested: Pendulum System