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8.11: End-of-Chapter Problem Set

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    End-of-Chapter Problem Set

    Structure and Setup

    1. Write the Excel formula for current \(I = V/R\) where V is in cell B1 and R is in B2. Use proper absolute references.
    2. Write the Excel formula for cable tension where mass is in B1, g is in B2, and angle is in the current row's column A. Use proper reference types.
    3. Identify all errors in this formula: =1000*9.81/(2*SIN(30))

    Parameter Sweeps

    1. Set up a tension vs. angle sweep from 10° to 80° in 5° increments for a 1000 kg load. Report tensions at 15°, 30°, 45°, and 75°. Generate a labeled scatter plot.
    2. Set up a power vs. voltage sweep from 1 V to 20 V for R = 470 Ω. Generate a labeled scatter plot. At what voltage does power exceed 0.5 W?
    3. A 9 V supply connects through a variable resistor swept from 100 Ω to 2000 Ω in 100 Ω steps. Compute current and power. Plot power vs. resistance. Interpret the shape.

    Graph Interpretation and Conditional Logic

    1. Describe three pieces of engineering information you would extract from a tension vs. angle graph. For each, state what design decision it would inform.
    2. A student's power graph has the y-axis starting at 0.20 W with data ranging 0.20–0.25 W. What problem does this create? What should be done?
    3. Write an Excel IF formula that flags a resistor as "UNSAFE" if power in cell C10 exceeds 0.25 W, and "SAFE" otherwise.
    4. [Cable Car Project prep] A cable car is 20 feet from the nearest tower. Using \(v = 0.3 d^{0.65}\), compute velocity by hand. Then write the Excel formula that computes it with d in cell D10.
    Answer Key - click to expand

    Structure and Setup

    1. Formula: =$B$1/$B$2
    2. Formula: =($B$1*$B$2)/(2*SIN(RADIANS(A10)))
    3. Errors:
      • The formula hardcodes \(1000\), \(9.81\), and \(30\) instead of referencing input cells.
      • Excel evaluates SIN(30) as 30 radians, not 30 degrees.
      • The angle should be written as SIN(RADIANS(30)), or better, should reference a cell containing the angle.
      • The formula cannot be copied correctly for a parameter sweep because the angle is fixed.

      A better single-case version would be =($B$1*$B$2)/(2*SIN(RADIANS($B$3))), assuming mass, gravity, and angle are in B1, B2, and B3.

    Parameter Sweeps

    1. Setup: Put the angles \(10, 15, 20, \ldots, 80\) in column A. Put \(m = 1000\) in B1 and \(g = 9.81\) in B2. In the first tension cell, use =($B$1*$B$2)/(2*SIN(RADIANS(A10))), then copy down.

      Selected results: At \(15°\), \(T \approx 18{,}952 \text{ N}\). At \(30°\), \(T = 9810 \text{ N}\). At \(45°\), \(T \approx 6937 \text{ N}\). At \(75°\), \(T \approx 5078 \text{ N}\).

      Graph: Use an XY scatter plot. Label the x-axis “Cable Angle (degrees)” and the y-axis “Tension (N).” A clear title is “Cable Tension vs. Cable Angle — 1000 kg Load.”

    2. Setup: Put \(R = 470\) in B1. Put voltage values \(1, 2, 3, \ldots, 20\) in column A. Use \(P = V^2/R\). In the first power cell, use =A10^2/$B$1, then copy down.

      The crossover voltage for \(0.5 \text{ W}\) is \(V = \sqrt{PR} = \sqrt{(0.5)(470)} \approx 15.33 \text{ V}\).

      Answer Power exceeds \(0.5 \text{ W}\) above about 15.3 V. In a 1 V increment table, the first listed voltage above the limit is 16 V.

      Graph: Use an XY scatter plot. Label the x-axis “Voltage (V)” and the y-axis “Power (W).” A clear title is “Power vs. Voltage — 470 Ω Resistor.”

    3. Setup: Put \(V = 9\) in B1. Put resistance values \(100, 200, 300, \ldots, 2000\) in column A. Use \(I = V/R\) and \(P = V^2/R\).

      Formulas: Current formula: =$B$1/A10. Power formula: =$B$1^2/A10.

      At \(R = 100 \ \Omega\), \(I = 0.09 \text{ A} = 90 \text{ mA}\) and \(P = 0.81 \text{ W}\). At \(R = 2000 \ \Omega\), \(I = 0.0045 \text{ A} = 4.5 \text{ mA}\) and \(P = 0.0405 \text{ W}\).

      Interpretation: Power decreases as resistance increases. The curve is nonlinear because resistance is in the denominator of \(P = V^2/R\). At low resistance, small resistance changes cause large power changes. At high resistance, the curve flattens.

    Graph Interpretation and Conditional Logic

    1. Three useful pieces of engineering information:
      • Maximum tension region: Low cable angles produce the highest tensions. This informs the minimum allowable cable angle.
      • Sensitivity region: The graph is steepest at small angles. This informs how tightly the installation angle must be controlled.
      • Rating crossover: A horizontal cable-rating line shows where the design becomes unsafe. This informs cable selection and safety factor decisions.
    2. A y-axis starting at \(0.20 \text{ W}\) can visually exaggerate small differences because the graph removes the zero baseline. A change from \(0.20\) to \(0.25 \text{ W}\) may look much larger than it really is.

      For introductory engineering communication, the y-axis should usually start at zero unless there is a clear reason to truncate it. If a truncated axis is used, it should be explicitly marked or explained.

    3. Formula: =IF(C10>0.25,"UNSAFE","SAFE")
    4. Hand calculation: \(v = 0.3(20)^{0.65} \approx 2.10\).

      Excel formula: =0.3*D10^0.65

      If the coefficient and exponent are stored in input cells, use absolute references. For example, if \(0.3\) is in B1 and \(0.65\) is in B2, use =$B$1*D10^$B$2.

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