Show that the moment of inertia of a uniform rectangular plate about any line in its plate is the same as that of four equal particles each of mass 1/12 of the whole at the angles, and a fifth of 2/3 of the whole mass at the centre.
Show,also, that the moment of inertia about any line perpendicular to the plane is the same for the disk and for these particles.
Applied Maths - Mechanics 58
2009-08-22 12:54 am
回答 (1)
2009-08-23 5:39 am
✔ 最佳答案
http://img233.imageshack.us/img233/6622/moi1i.jpg圖片參考：http://img233.imageshack.us/img233/6622/moi1i.jpg
http://img216.imageshack.us/img216/586/moi2u.jpg
Distance2 of G from O = (x + W/2)2 + (y + H/2)2
Moment of inertia of plate about G = M[(x + W/2)2 + (y + H/2)2] + (M/12)(W2 + H2)
Moment of inertia of the particle system about G is (2M/3)[(x + W/2)2 + (y + H/2)2] +
(M/12){[x2 + y2] + [x2 + (y + H)2] + [(x + W)2 + y2] + [(x + W)2 + (y + H)2]}
= (2M/3)[(x + W/2)2 + (y + H/2)2] +
(M/12)(x2 + y2 + x2 + y2 + 2Hy + H2 + x2 + 2Wx + W2 + y2 + x2 + 2Wx + W2 + y2 + 2Hy + H2)
= (2M/3)[(x + W/2)2 + (y + H/2)2] + (M/12)(4x2 + 4y2 + 4Hy + 2H2 + 4Wx + 2W2)
= (2M/3)[(x + W/2)2 + (y + H/2)2] + (M/12)(4x2 + 4Wx + W2 + 4y2 + 4Hy + H2 + H2 + W2)
= (2M/3)[(x + W/2)2 + (y + H/2)2] + (M/12)[(2x + W)2 + (2y + H)2 + H2 + W2]
= (2M/3)[(x + W/2)2 + (y + H/2)2] + (M/3)[(x + W/2)2 + (y + H/2)2] + (M/12)(H2 + W2)
= M[(x + W/2)2 + (y + H/2)2] + (M/12)(H2 + W2)
= Moment of inertia of plate about G
圖片參考：http://img216.imageshack.us/img216/586/moi2u.jpg
2009-08-22 21:40:58 補充：
The calculation was not shown correctly for the first part, please see
http://img233.imageshack.us/img233/6622/moi1i.jpg
收錄日期: 2021-04-23 23:24:58
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