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Zorluk: OrtaLevers and Fulcrums

A shipyard rigger needs to lift a 200-pound200\text{-pound} machinery casing. The rigger places one end of a rigid 60-inch60\text{-inch} steel pry bar under the casing and positions a fulcrum beneath the bar, creating a first-class lever. If the rigger applies a downward effort force of 50 pounds50\text{ pounds} at the opposite end of the bar, how far from the casing must the fulcrum be placed to perfectly balance the load?

  1. A
    15 inches15\text{ inches}
  2. 12 inches12\text{ inches}Cevap
  3. C
    20 inches20\text{ inches}
  4. D
    48 inches48\text{ inches}

Cevap

12 inches12\text{ inches}
To balance a first-class lever, the torque generated by the load must exactly equal the torque generated by the effort force. Setting the fulcrum 12 inches12\text{ inches} from the 200-pound200\text{-pound} load leaves 48 inches48\text{ inches} of the bar for the effort arm (60−12=4860 - 12 = 48). The torque on the load side is 200 lbs×12 inches=2400 inch-pounds200\text{ lbs} \times 12\text{ inches} = 2400\text{ inch-pounds}, which perfectly matches the effort torque of 50 lbs×48 inches=2400 inch-pounds50\text{ lbs} \times 48\text{ inches} = 2400\text{ inch-pounds}.

Adım Adım Çözüm

1
Set up the torque balance equation for a first-class lever.
Fload×dload=Feffort×deffortF_{\text{load}} \times d_{\text{load}} = F_{\text{effort}} \times d_{\text{effort}}
For the lever to balance, the rotational force (torque) applied by the load must equal the torque applied by the effort.
2
Define the lever arms in terms of the unknown distance from the load to the fulcrum (xx).
dload=xd_{\text{load}} = x
deffort=60−xd_{\text{effort}} = 60 - x
The fulcrum splits the total 60-inch60\text{-inch} bar into two segments. If one side is xx, the remaining length is 60−x60 - x.
3
Substitute the known values into the equation and solve for xx.
200⋅x=50⋅(60−x)200 \cdot x = 50 \cdot (60 - x)
200x=3000−50x200x = 3000 - 50x
250x=3000250x = 3000
x=12 inchesx = 12\text{ inches}
Using basic algebra isolates xx, revealing the exact fulcrum placement required for balance.

Anahtar Kavram

Torque balance and lever arms in a first-class lever
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Konular

Levers and Fulcrums

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