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Difficulty: Very hardInclined Planes

Calculate the total effort force required for each scenario and arrange the following inclined plane configurations in order from the GREATEST required effort force to the LEAST required effort force.

  1. 1A 240 lb240\text{ lb} crate moved up a 12 ft12\text{ ft} ramp to a height of 3 ft3\text{ ft}, with a constant friction force of 30 lb30\text{ lb}.
  2. 2A 160 lb160\text{ lb} crate moved up an 8 ft8\text{ ft} ramp to a height of 4 ft4\text{ ft}, assuming perfectly frictionless conditions.
  3. 3A 300 lb300\text{ lb} crate moved up a 15 ft15\text{ ft} ramp to a height of 3 ft3\text{ ft}, with a constant friction force of 15 lb15\text{ lb}.
  4. 4A 450 lb450\text{ lb} crate moved up a 20 ft20\text{ ft} ramp to a height of 2 ft2\text{ ft}, with a constant friction force of 25 lb25\text{ lb}.

Answer

The correct sequence from greatest to least effort force is: 240 lb crate (90 lb effort), 160 lb crate (80 lb effort), 300 lb crate (75 lb effort), and 450 lb crate (70 lb effort).
The total effort force required is the sum of the ideal effort (weight divided by IMA) and the friction force. For the 240 lb crate, the total effort is 90 lb. For the 160 lb crate, it is 80 lb. For the 300 lb crate, it is 75 lb. For the 450 lb crate, it is 70 lb. Arranging these from greatest to least yields the sequence of 90 lb, 80 lb, 75 lb, and 70 lb.

Step-by-Step Solution

1
Calculate the Ideal Mechanical Advantage (IMA) for each ramp.
Item 1: 5. Item 2: 2. Item 3: 10. Item 4: 4.
IMA is determined by the ratio of the ramp's length to its vertical height (IMA=LhIMA = \frac{L}{h}).
2
Determine the ideal effort force required to move each load without friction.
Item 1: 60 lb. Item 2: 80 lb. Item 3: 45 lb. Item 4: 60 lb.
The ideal effort force is the weight of the load divided by the IMA (Fideal=WIMAF_{\text{ideal}} = \frac{W}{IMA}).
3
Add the constant friction force to the ideal effort force to find the total required effort.
Item 1: 75 lb. Item 2: 80 lb. Item 3: 70 lb. Item 4: 90 lb.
Friction acts against the movement up the ramp, so it must be added to the ideal effort force to find the true required effort.
4
Arrange the final total effort values in descending order.
90 lb, 80 lb, 75 lb, 70 lb.
The question asks for the order from greatest required effort force to least.

Key Concept

Calculating total required effort force on an inclined plane by combining Ideal Mechanical Advantage (IMA) and frictional forces.
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Topics

Inclined Planes

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