標題:
Applied Maths - Mechanics 58
發問:
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... 顯示更多 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.
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最佳解答:
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
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