Hydraulic Lift System Lecture Notes

Introduction to Hydraulic Lift System

• Problem involving a downward force of 100 newtons applied to a small piston with a 50 cm diameter.
• Objective: Find the upward force exerted by a large piston with a 2-meter diameter.

Key Concepts

Pascal's Law

• Pressure applied to an enclosed fluid is transmitted undiminished throughout the fluid.
• Pressure on small piston (P1) = Pressure on large piston (P2).

Pressure and Force Relationship

• Pressure (P): Force (F) divided by Area (A)

• For the output force (F2), solve using the equation:

  [ F2 = F1 \times \frac{A2}{A1} ]

Circular Pistons

• Area of a piston: ( \pi r^2 )
• Radius is half of the diameter.
• Convert cm to meters for calculations.

Calculations

Part A: Upward Force Calculation

• Given:
  • F1 = 100 N
  • Diameter of large piston = 2m, radius = 1m
  • Diameter of small piston = 50cm, radius = 0.25m
• Calculation:
  • Cancel ( \pi )
  • ( \left(\frac{1^2}{0.25^2}\right) = 16 )
  • F2 = 100 N ( \times 16 = 1600 N )

Part B: Mechanical Advantage

• Definition: Ratio of output force (F2) to input force (F1)
• Mechanical Advantage = 16
• Example: An input force of 1000 N results in an output force of 16000 N.

Part C: Piston Displacement

• Hydraulic Lift System Volume Conservation:

  • Volume change in small piston = Volume change in large piston.
  • V1 = V2 (A1 ( \times ) D1 = A2 ( \times ) D2)

• Calculation:

  • Input force (F1) of 100 N displaces small piston by 2 m.
  • D2 = 0.125 m for large piston.

Energy Conservation

• Input work = Output work
• ( F1 \times D1 = F2 \times D2 )
• Example: 100 N ( \times 2 m = 1600 N \times 0.125 m )

Applications

• Example with a 1500 kg car on a large piston with a 4m radius.
  • Minimum F2 for lifting at a constant speed = mg = 14,700 N.

Mechanical Advantage Example

• Given mechanical advantage of 20, find F1:
  • F1 = 14,700 N / 20 = 735 N

Small Piston Radius Calculation

• Use Pascal's Law and cross-multiplying:
  • Mechanical advantage relates forces and areas.
  • R1 = 0.894 m

Pressure Calculation

• Pressure = Force / Area
• Large piston pressure = 14,700 N / (( \pi \times 16 )) = 292 Pascals

Summary of Key Formulas

• Mechanical Advantage: ( \frac{F2}{F1} = \frac{A2}{A1} = \frac{D1}{D2} )
• Force Output: ( F2 = F1 \times \frac{A2}{A1} )
• Area (circular): ( \pi r^2 )
• Input Work = Output Work: ( F1 \times D1 = F2 \times D2 )
• Volume Conservation: ( A1 \times D1 = A2 \times D2 )
• Pressure Consistency: P1 = P2

Conclusion

• Hydraulic systems are force-multiplying devices.
• Multiple approaches to solve problems, including energy conservation and Pascal's Law.
• Important to understand the relationship between force, area, and displacement in hydraulic systems.