球体体积知道,高度分10000等份,介质为氨,每等份的质量是多少,如何计算?
球体上下半球对称,只需计算上半球,如图。
第1层球冠可看作上底半径为0的球台;以下各层都是球台。层高 h 为 R/5000。
L1^2 = R^2 - ( R - R/5000 )^2
= 2R^2/5000 - ( R/5000 )^2 = R^2( 10000 - 1 )/5000^2;
L2^2 = R^2 - ( R - 2R/5000 )^2
= 2 * 2R^2/5000 - ( 2R/5000 )^2 = R^2( 2 * 10000 - 2^2 )/5000^2;
L3^2 = R^2 - ( R - 3R/5000 )^2
= 2 * 3R^2/5000 - ( 3R/5000 )^2 = R^2( 3 * 10000 - 3^2 )/5000^2;
……
第 n 层上底半径平方 (Ln-1)^2 = R^2 - [ R - (n-1)R/5000 )^2
= 2 * (n-1)R^2/5000 - [ (n-1)R/5000 ]^2 = R^2[ 10000(n-1) - (n-1)^2 ]/5000^2;
第 n 层下底半径 Ln = R^2 - [ R - nR/5000 )^2
= 2nR^2/5000 - [ nR/5000 ]^2 = R^2[ 10000n - n^2 )/5000^2;
第 n 层球台体积 Vn = (πh/6){ 3(Ln-1)^2 + 3(Ln)^2 + h^2 }
= (π/6)( R/5000 ){ 3R^2[ 10000(n-1) - (n-1)^2 ]/5000^2
+ 3R^2[ 10000n - n^2 ]/5000^2 + ( R/5000 )^2 }
= (πR^3/6){ 60000n - 30000 - 6n^2 + 6n - 3 + 1 }/5000^3
= (πR^3/6)( 60006n - 30002 - 6n^2 )/5000^3 ①
球体积 V = (4/3)πR^3,R^3 = 3V/(4π),代入①
Vn = (V/8)( 60006n - 30002 - 6n^2 )/5000^3;
氨的标准密度为 0.77,
故第n层氨的质量 mn = 0.77Vn = (77V/800)( 60006n - 30002 - 6n^2 )/5000^3 。