Applied Maths - Mechanics 5

還不明白的薑

2009-07-27 5:51 am

A toy towel is constructed of thin uniform sheet metal and consists of a right circular cylinder, with open ends, of height h and radius r, where h>r, covered by a hemispherical cap od radius r. Show that, for such a tower, the centre of gravity is at a distance x from the ground where

x = 0.5(h+r) .

The tower is suspended from a string which is attached to a point on the rim of the hemispherical cap. Show that, when the tower hangs in equilibrium, the axis of symmetry of the tower is inclined at an angle @ to the horizontal, where

tan@ = (h-r)/2r

回答 (1)

用戶9112

2009-07-27 8:32 am

✔ 最佳答案

The centre of gravity of a hemispherical shell is located at 1/2*radius from the circular end of the shell along the axis of symmetry.
The centre of gravity of a cylindrical shell is located at 1/2*height from one of the circular end of the shell along the central axis.
Mass of hemispherical shell : Mass of cylindrical shell = 2πr2 : 2πrh = r:h
Total mass : mass of cylindrical shell = r+h : h
Centre of gravity of the hemispherical shell from the base = h + r/2
Let the centre of gravity of the whole object be at x from the base.
Take moment from the centre of the base, (Distance x Mass)
(h + r/2)(r) + (h/2)(h) = (x)(r + h)
rh + r2/2 + h2/h = x(r + h)
(1/2)(r + h)2 = x(r + h)
x = 0.5(r + h)
The centre of mass when measured from the plane of the hemispherical cap is h – x
Refer to the diagram,
http://img512.imageshack.us/img512/5968/towerl.jpg
tan@ = (h – x) / r = (h – 0.5h – 0.5r)/r = (h – r)/2r

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