Applied Maths - Mechanics 7

還不明白的薑

2009-07-27 9:25 pm

A uniform solid peg consists of a right circular cone of height h and base radius r joined at the base to a hemisphere also of radius r. Show that the peg will remains in equilibrium when it is placed with the surface of the cone in contact with a horizontal plane. You may assume that the centre of gravity of the hemisphere is situated at a distance 3r/8 from its base.

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用戶9112

2009-07-28 7:23 am

✔ 最佳答案

Use "Applied Maths - Mechanics 4" as reference, the centre of mass of a Cone + Hemisphere is located at (3r2 – h2)/[4(h + 2r)] where density A = density B, and the direction towards the hemisphere is taken as positive.
Refer to the diagram, as far as the centre of mass does not go beyond u into the hemisphere, the solid can rest of the cone surface.
http://img89.imageshack.us/img89/6367/icecreamx.jpg
Now tan x = r/h = u/r so u = r2/h.
Compute u - (3r2 – h2)/[4(h + 2r)]
= r2/h - (3r2 – h2)/[4(h + 2r)] = (8r3 + hr2 + h3) / [4h(h + 2r)] > 0
Hence the centre of mass does not go beyond u and so the cone can rest on the cone surface on a horizontal plane.

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