PLTW Principles of Engineering: Static Equilibrium, Wheel and Axle, and Pulley Calculations

Static Equilibrium and Lever Calculations

• Objective: Calculate the required distance from the fulcrum to hold tweezers to avoid damaging a splinter using static equilibrium principles.
• Governing Formulas:
  • Moment equation: $M = F \times D$
  • Static equilibrium moment condition: $M_E = M_R$
    • $M_E$ = Moment of Effort
    • $M_R$ = Moment of Resistance
• Substitution and Solution:
  • Equating effort moment and resistance moment:         $1 \times 4 = D_E \times 5$
  • Simplifying:         $4 = D_E \times 5$
  • Solving for Effort Distance ($D_E$):         $D_E = \frac{4}{5} = 0.8$
• Final Answer:
  • $0.8\,\text{in}$ From Fulcrum

Wheel and Axle Linear Distance Calculations

• Objective: Calculate the linear distance traveled in one full revolution of a $36\,\text{in}$ diameter wheel.
• Given Parameters:
  • Wheel diameter ($d$): $36\,\text{in}$
  • Wheel radius ($r$): $\frac{36}{2} = 18\,\text{in}$
• Governing Formulas:
  • Circumference of a circle: $C = 2\pi r$
  • Linear distance relationship: $d_{linear} = 2r$
• Substitution and Solution:
  • Calculating circumference using radius:         $C = 2 \times \pi \times 18$
  • Numerical approximation:         $C = 113.1\,\text{in}$
  • Exact value in terms of $\pi$:         $C = 36\pi$
• Final Answer:
  • $113.1\,\text{in}$ or $36\pi\,\text{in}$

Pulley System Calculations: Rooftop Material Lift

• System Context:
  • A construction crew lifts approximately $560\,\text{lb}$ of material several times during a day from a flatbed truck to a $32\,\text{ft}$ rooftop.
  • A block and tackle system with $50\,\text{lb}$ of effort force ($F_E$) is designed to lift the materials.
• Required Actual Mechanical Advantage (AMA):
  • Governing Formulas:
    • Relationship between Ideal Mechanical Advantage and Actual Mechanical Advantage: $IMA = AMA$
    • Actual Mechanical Advantage formula: $AMA = \frac{F_R}{F_E}$
      • $F_R$ = Resistance Force ($560\,\text{lb}$)
      • $F_E$ = Effort Force ($50\,\text{lb}$)
  • Substitution and Solution:         $AMA = \frac{560}{50} = 11.2$
  • Final Answer:         $AMA = 11.2$
• Required Supporting Strands:
  • Governing Concept:
    • $IMA = AMA$
    • The actual mechanical advantage dictates the minimum mechanical ratio required: $\frac{560}{11.2} = 50$
    • Number of supporting strands ($D_E / D_R$ ratio) must be rounded up to the nearest whole strand capable of supporting the system load.
  • Substitution and Solution:         $\text{Strands} = 12$
  • Final Answer:         $12\text{ Strands}$

Pulley System Calculations: Manufacturing Facility Metal Lathe

• System Context:
  • A block and tackle system with $9$ supporting strands is used to lift a metal lathe in a manufacturing facility.
  • The motor being used to wind the cable in the pulley system can provide $100\,\text{lb}$ of force.
• Mechanical Advantage Determination:
  • Governing Concept:
    • For a block and tackle pulley system, the mechanical advantage (Ideal Mechanical Advantage) is equal to the number of supporting strands supporting the load.
  • System Value:
    • Number of supporting strands = $9$
    • $FMA = 9$
  • Final Answer:
    • $\text{mechanical advantage} = 9$