Question

how we can prove arcmdies principal? ​

Answer 1

How do you prove Archimedes principle?

The overall force F an object submerged in liquid experiences is,

F = ∮S(-)ρgzds, S is the submerged boundary of the object.

∵ ds = i(i⋅ds) + j(j⋅ds) + k(k⋅ds), i, j and k are unit vector in +x, +y and +z direction respectively.

∴ F = -ρg∮Sz(i(i⋅ds) + j(j⋅ds) + k(k⋅ds)) = (-ρg∮S(zi)⋅ds)i + (-ρg∮S(zj)⋅ds)j + (-ρg∮S(zk)⋅ds)k = (-ρg∮V∇⋅ (zi)⋅dv)i + (-ρg∮V∇⋅ (zj)⋅dv)j + (-ρg∮V∇⋅ (zk)⋅dv)k

∵ ∇⋅ (zi) = 0, ∇⋅ (zj) = 0, ∇⋅ (zk) = 1

∴ F = (-ρg∮Vdv)k = -ρgVk

The magnitude of the overall force (buoyant force) is ρgV, and the direction is -k.

--JF Hu

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Answer 2

it's already been proved..you can design a submarine and get it proved according to the law