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Zorluk: Çok zorScrews and Threads

A machinist tests two threaded shafts being driven into fixed, stationary bases. Shaft A is equipped with a 14-inch14\text{-inch} handle and standard right-hand threads at 44 threads per inch (TPI). Shaft B is equipped with a 7-inch7\text{-inch} handle and left-hand threads at 88 TPI. From a top-down view, the machinist turns the handle of both shafts exactly one full revolution clockwise. Assuming no friction, which of the following statements correctly describes the linear movement of the shafts and compares their Ideal Mechanical Advantage (IMA)?

  1. Shaft A moves exactly 0.250.25 inches downward into the base, Shaft B moves exactly 0.1250.125 inches upward out of the base, and both shafts possess the exact same IMA.Cevap
  2. B
    Both shafts move downward into their respective bases, but Shaft B possesses twice the IMA of Shaft A due to its finer thread pitch.
  3. C
    Shaft A moves exactly 0.250.25 inches upward out of the base, Shaft B moves exactly 0.1250.125 inches downward into the base, and both shafts possess the exact same IMA.
  4. D
    Shaft A moves exactly 0.250.25 inches downward into the base, Shaft B moves exactly 0.1250.125 inches upward out of the base, and Shaft A possesses four times the IMA of Shaft B.

Cevap

Shaft A moves 0.250.25 inches downward, Shaft B moves 0.1250.125 inches upward, and both shafts have the exact same Ideal Mechanical Advantage.
The correct answer accurately calculates that a clockwise turn drives a right-hand thread downward by its pitch (1/4=0.251/4 = 0.25 inches) and a left-hand thread upward by its pitch (1/8=0.1251/8 = 0.125 inches). Furthermore, it recognizes that Ideal Mechanical Advantage (IMA) is the effort circumference (2πR2\pi R) multiplied by the TPI. The larger 14-inch14\text{-inch} handle on Shaft A exactly compensates for its coarser 44 TPI pitch, resulting in an identical IMA (112π112\pi) for both systems.

Adım Adım Çözüm

1
Determine the direction of linear movement based on thread handedness.
Shaft A moves downward; Shaft B moves upward.
By convention, turning a standard right-hand thread clockwise drives it inward (downward into the base). Turning a left-hand thread clockwise drives it outward (upward out of the base).
2
Calculate the linear travel distance per revolution (pitch) for each shaft.
Shaft A travels 0.250.25 inches; Shaft B travels 0.1250.125 inches.
Pitch is the reciprocal of Threads Per Inch (TPI). For Shaft A: 1/4=0.251 / 4 = 0.25 inches. For Shaft B: 1/8=0.1251 / 8 = 0.125 inches.
3
Calculate the Ideal Mechanical Advantage (IMA) for Shaft A.
IMAA=112π\text{IMA}_A = 112\pi
The IMA of a screw is the ratio of effort distance to load distance per revolution: (2⋅π⋅R)/Pitch(2 \cdot \pi \cdot R) / \text{Pitch}. For A: (2⋅π⋅14)/0.25=28π/0.25=112π(2 \cdot \pi \cdot 14) / 0.25 = 28\pi / 0.25 = 112\pi.
4
Calculate the Ideal Mechanical Advantage (IMA) for Shaft B.
IMAB=112π\text{IMA}_B = 112\pi
Applying the same formula for B: (2⋅π⋅7)/0.125=14π/0.125=112π(2 \cdot \pi \cdot 7) / 0.125 = 14\pi / 0.125 = 112\pi. The doubling of the TPI exactly offsets the halving of the handle length, yielding identical IMA.

Anahtar Kavram

The IMA of a screw depends on both the length of the lever arm (handle) and the pitch of the threads, and thread handedness determines linear direction relative to rotation.
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Screws and Threads

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