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Difficulty: HardWheel and Axle

Match each wheel and axle mechanism to its corresponding force-distance trade-off profile and Ideal Mechanical Advantage (IMA).

  • A user turns a 1.5-inch1.5\text{-inch} diameter screwdriver handle to drive a screw using the 0.25-inch0.25\text{-inch} diameter shaft.Acts as a force multiplier with an IMA of 66, applying greater torque to the fastener.
  • A bicycle chain drives a 3-inch3\text{-inch} diameter rear sprocket, which is rigidly attached to a 27-inch27\text{-inch} diameter rear tire.Acts as a distance multiplier with an IMA of 19\frac{1}{9}, increasing the linear travel per revolution.
  • A driver operates a 14-inch14\text{-inch} diameter steering wheel to rotate a 1-inch1\text{-inch} diameter central steering column.Acts as a force multiplier with an IMA of 1414, providing the operator with significant leverage.
  • A motor turns a 0.5-inch0.5\text{-inch} diameter central drive shaft that spins a blade assembly spanning a 50-inch50\text{-inch} diameter.Acts as a speed multiplier with an IMA of 1100\frac{1}{100}, generating high velocity at the blade tips.

Answer

Screwdriver maps to IMA 6 (force multiplier). Bicycle maps to IMA 1/9 (distance multiplier). Steering wheel maps to IMA 14 (force multiplier). Ceiling fan maps to IMA 1/100 (speed multiplier).
Each mechanism is correctly paired by identifying where the effort is applied (wheel or axle) and calculating the ratio of the effort diameter to the load diameter. This ratio directly dictates whether the machine sacrifices distance for force (IMA > 1) or sacrifices force for distance/speed (IMA < 1).

Step-by-Step Solution

1
Identify the effort and load components for each mechanism.
Determine which part acts as the 'wheel' (larger radius) and which acts as the 'axle' (smaller radius), and definitively locate where the input effort is applied.
The formula for Ideal Mechanical Advantage requires knowing whether the input is applied to the wheel or the axle.
2
Calculate the Ideal Mechanical Advantage (IMA) for each item.
Using the formula IMA = (Diameter of Effort) / (Diameter of Load). Screwdriver: 1.5/0.25 = 6. Bicycle: 3/27 = 1/9. Steering Wheel: 14/1 = 14. Ceiling Fan: 0.5/50 = 1/100.
This mathematical ratio represents the mechanical trade-off of the machine.
3
Classify each mechanism based on its IMA value.
IMA > 1 (Screwdriver, Steering Wheel) means the machine multiplies force. IMA < 1 (Bicycle, Ceiling Fan) means the machine multiplies distance or speed.
A machine cannot multiply both force and distance simultaneously due to the conservation of energy.

Key Concept

Ideal Mechanical Advantage (IMA) of a Wheel and Axle
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Topics

Wheel and Axle

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