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phantomvommand

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Thread moved from the technical forums to the schoolwork forums

**TL;DR Summary:**A thin piece of spaghetti of diameter d is balanced horizontally from its middle.

It can have a length ℓ ≫ d before it snaps under its own weight. How does ℓ scale with d?

Let the spaghetti rod have density ρ, and consider ONLY its right half.

(1) The spaghetti rod experiences some horizontal stress, let it be ##\sigma##. By taking moments about the pivot point, moment due to weight must be balanced by moments due to the horizontal stress. Therefore,

##\rho d^2 l (l) \sim \sigma d^2 (d)##, so ##\sigma \sim \rho \frac {l^2} {d}##.

This agrees with the textbook solution.

However, the text makes an additional remark that:

(2) There must be a vertical normal force (at the pivot) ##F \sim ρd^2ℓ## to balance the weight. This vertical force is transmitted through the rod by a shear stress (i.e. an internal force per area, perpendicular to the rod) of order ##σ_v## ∼ F/A ∼ ρℓ.

It then compares the 2 stresses, and concludes that: Horizontal stress is much greater than ##σ_v##, because of the miserably small lever arm, which is why thin rods usually break by snapping, not by shearing or pulling apart.

How did it reach this conclusion? I get that horizontal stress is much larger than vertical stress, but how does that lead to the claim that thin rods are more likely to break by snapping?

Doesn't it depends on the maximum horizontal and vertical stress for the material?

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