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Difficulty: MediumWheel and Axle

Match each real-world wheel and axle application to its corresponding Ideal Mechanical Advantage (IMA). Remember to identify whether the input force is applied to the wheel or the axle.

  • A hand winch where the operator turns a 15-inch15\text{-inch} crank arm to wind a cable around a 3-inch3\text{-inch} radius drum.IMA=5\text{IMA} = 5
  • A bicycle rear wheel where the chain drives a 2-inch2\text{-inch} radius sprocket attached to a 14-inch14\text{-inch} radius tire.IMA=17\text{IMA} = \frac{1}{7}
  • A steering mechanism where the driver turns a 12-inch12\text{-inch} radius steering wheel connected to a 2-inch2\text{-inch} radius steering column.IMA=6\text{IMA} = 6
  • A ceiling fan where the internal motor rotates a 1-inch1\text{-inch} radius shaft that turns blades with a 20-inch20\text{-inch} radius.IMA=120\text{IMA} = \frac{1}{20}

Answer

Hand winch: IMA = 5; Bicycle rear wheel: IMA = 1/7; Steering mechanism: IMA = 6; Ceiling fan: IMA = 1/20.
Each real-world application must be evaluated by identifying the radius where the input force is applied and dividing it by the radius where the output force is exerted.

Step-by-Step Solution

1
Identify the input and output components for each mechanism.
Determine if the input force is applied to the larger radius (wheel) or the smaller radius (axle).
IMA depends on where the input force is applied.
2
Recall the formula for Ideal Mechanical Advantage (IMA) of a wheel and axle.
IMA=Radius of InputRadius of Output\text{IMA} = \frac{\text{Radius of Input}}{\text{Radius of Output}}.
This formula provides the theoretical mechanical trade-off of the system.
3
Calculate IMA for mechanisms acting as force multipliers (input on the larger wheel).
Winch: 153=5\frac{15}{3} = 5. Steering mechanism: 122=6\frac{12}{2} = 6.
These devices decrease the required input force but require it to be applied over a greater distance, resulting in an IMA > 1.
4
Calculate IMA for mechanisms acting as speed or distance multipliers (input on the smaller axle).
Bicycle: 214=17\frac{2}{14} = \frac{1}{7}. Ceiling fan: 120\frac{1}{20}.
These devices increase output speed and distance, but require a larger input force, resulting in an IMA < 1.

Key Concept

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

Wheel and Axle

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