Applied Maths - Mechanics 4

還不明白的薑

2009-07-27 5:46 am

Find the position of the centre of mass of a solid consisting of a right circular cone and a hemisphere, where the height and the radius of the cone are h and r respectively, the radius of the hemisphere is also r. Densities of the cone and the hemisphere are A and B respectively.

更新1:

ANS : (9Br^2 - 4Ah^2)/12(2Br + Ah) from the plane face

回答 (1)

用戶9112

2009-07-28 6:58 am

✔ 最佳答案

The centre of mass of a solid cone is located at 1/4 of the height from the base.
The centre of mass of a solid hemisphere is located at 3/8 of the radius from the circular base of the hemisphere.
Mass of the circular cone = (A)(1/3)πr2h
Mass of the hemisphere = (B)(2/3)πr3
Define the direction towards the hemisphere is positive.
Let the centre of mass be located at x.
Take mass moment about the centre of the interfacing circle along the axis of symmetry =
(B)(2/3)πr3(3r/8) – (A)(1/3)πr2h(h/4) = [(A)(1/3)πr2h + (B)(2/3)πr3]x
Eliminate πr2/3 from all terms,
2Br(3r/8) – Ah(h/4) = (Ah + 2Br)x
(3Br2 – Ah2) / 4 = (Ah + 2Br)x
x = (3Br2 – Ah2) / 4(Ah + 2Br)
(*The provided answer is wrong which is based upon a centre of gravity of h/3 for a cone.*)

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