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Difficulty: HardInclined Planes

A logistics crew needs to load an 800 lb800\text{ lb} supply crate onto a transport plane whose cargo deck is 6 ft6\text{ ft} above the tarmac. The crew can apply a maximum combined effort force of 200 lb200\text{ lb}. Assuming friction is negligible, what is the minimum ramp length required, and what is the total work done to lift the crate?

  1. Minimum ramp length of 24 ft24\text{ ft}; total work of 4,800 ft-lb4,800\text{ ft-lb}Answer
  2. B
    Minimum ramp length of 24 ft24\text{ ft}; total work of 1,200 ft-lb1,200\text{ ft-lb}
  3. C
    Minimum ramp length of 4 ft4\text{ ft}; total work of 4,800 ft-lb4,800\text{ ft-lb}
  4. D
    Minimum ramp length of 24 ft24\text{ ft}; total work of 19,200 ft-lb19,200\text{ ft-lb}

Answer

Minimum ramp length of 24 ft24\text{ ft}; total work of 4,800 ft-lb4,800\text{ ft-lb}
The Ideal Mechanical Advantage required is the load divided by the effort (800 / 200 = 4). Since IMA also equals length divided by height, the length must be 4 times the height (4 * 6 = 24 ft). Total work is the force times the distance it moves in the direction of the force, which can be calculated as Load * Height (800 * 6 = 4,800 ft-lb) or Effort * Length (200 * 24 = 4,800 ft-lb).

Step-by-Step Solution

1
Calculate the required Ideal Mechanical Advantage (IMA).
IMA=LoadEffort=800 lb200 lb=4\text{IMA} = \frac{\text{Load}}{\text{Effort}} = \frac{800\text{ lb}}{200\text{ lb}} = 4
The IMA tells us how much the machine (ramp) multiplies the applied force.
2
Calculate the minimum ramp length using the IMA and the given height.
Length=IMA×Height=4×6 ft=24 ft\text{Length} = \text{IMA} \times \text{Height} = 4 \times 6\text{ ft} = 24\text{ ft}
For an inclined plane, IMA=LengthHeight\text{IMA} = \frac{\text{Length}}{\text{Height}}. Rearranging gives Length=IMA×Height\text{Length} = \text{IMA} \times \text{Height}.
3
Calculate the total work done.
Work=Load×Height=800 lb×6 ft=4,800 ft-lb\text{Work} = \text{Load} \times \text{Height} = 800\text{ lb} \times 6\text{ ft} = 4,800\text{ ft-lb}
Work can be calculated as the vertical work required to lift the load. Alternatively, Work=Effort×Length=200 lb×24 ft=4,800 ft-lb\text{Work} = \text{Effort} \times \text{Length} = 200\text{ lb} \times 24\text{ ft} = 4,800\text{ ft-lb}.

Key Concept

The Ideal Mechanical Advantage of an inclined plane dictates its proportions (length vs. height) to reduce effort force, while the total work done remains constant (conservation of energy).
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Topics

Inclined Planes

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