球体[tex=8.286x1.286]QwY3CbnOdl+ukx2Eamho1GB4Xbk0HfxM+Hh/UALYDYAYK/LNG65QZxjuijAJcJbo[/tex]内，各点处的密度的大小等于该点到坐标原点的距离的平方，求该球体的重心。

2022-11-03

球体[tex=8.286x1.286]QwY3CbnOdl+ukx2Eamho1GB4Xbk0HfxM+Hh/UALYDYAYK/LNG65QZxjuijAJcJbo[/tex]内，各点处的密度的大小等于该点到坐标原点的距离的平方，求该球体的重心。

答案：

解：由于此球体关于[tex=1.857x1.286]1Yluhflx57gm9qrRVV1mkg==[/tex]，[tex=1.786x1.286]j94FgCsn47moOGKYTFDU/A==[/tex]坐标面对称，所以[tex=4.143x1.286]s6KXmC4v6t7+qIZ2Bi1qoxL4A8zB6S2/VKdpQmoED08=[/tex]，下面求[tex=0.571x1.286]RHvcsiG+OITcl6gcBst+og==[/tex]。[tex=0.714x1.286]1YkIdjxXLHdjdjLEO+eusQ==[/tex]为球体，用球坐标计算三重积分，将球坐标代入球面方程，得[tex=7.071x1.286]nq4uRkaQs+EIyg+hv21wA7JKLotaOqNAcD7OBKT2iggqPfAvDss/wU4pV6Ibc9f0[/tex]，又易见[tex=4.643x1.286]whEZdHAFuhT88iWBX/aj58wJ9cuziA5H2DvriayiP7Y=[/tex]，[tex=4.5x1.786]IURGdIB5fISq0xcMGb+Cwu3Ti63jGuXq8jPn3vCcKsMxsLD0DULUtBxGkbltXEJY[/tex]所以[tex=19.286x2.5]7gb+n+3TX+LvRI8M7TQAX47z3TU8bUg89mS3L6Hr5IGUYagwsuxG+N6SC6eMQp+ayNST5X2dNxUADukNrWRXM/Kf+h62trIWoySzDpiUaj4Px8tNp9QQ1mFfo/fGNisla5Vc/nl6nPxNP522LrQzhtQXMbU79WQcLMbAQvvQKO4hXLcHxv8EIsaO7AX3iQHd[/tex][tex=14.786x2.5]hfbN/DvAm9Wjp9EkBCeYLdXqXNlpiwQp8l1BbkB3PpQUFGTp5qLhX0xWBlts1TWL5Xr7UY73y9ntdafEO2FEv+1PC7JOAPHjV0gZBa+BO2+ZJHpIv2e03WdC77TpK1hDlMXE20hbNel7Yi1np1ZTabV7LcPkdzOmHvRwKkUBeHV2WTShgHYp8nUNGGOUla2e[/tex][tex=14.643x2.5]/B3hhxcc81/Sb4YY+NY+K8ch93i5/OowRYRJUkX8gy7dDtdDaIuY3iR1QOoKn8GmDIe/V24xRBIYPb1j/CksKhST5yJ6NxDxX9aR1zxS+Elpz0yA7i7zR6aB/7zOt/JVJjXFpMXSsNDTRpEyKBAH1Q==[/tex][tex=15.214x2.714]/B3hhxcc81/Sb4YY+NY+K8ch93i5/OowRYRJUkX8gy4FQDh7D9xEpwWsiWASIAu3+O6EH1OHjahLSwa/0iwTWBjV8LjpYQVRJHui3jWX9Z0hmYR6uwoDn9reZJlXOUyCd5t13ztSKnInfQuMuvIQkqVqa6ADfk+boGA488DJ/0JsdZFkcYZwI3L40H+L/DMv[/tex]静矩[tex=28.0x11.786]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[/tex][tex=0.714x1.286]Mjp1ERIg12NQkOrp1BseMg==[/tex][tex=6.571x2.0]noSJcKBLQiVP68rsCcl+lshwiviWgSWRM19RYWEPzV1NjRqjoogshnrM1bBWG8fI[/tex]，故该球体的重心坐标为[tex=4.571x2.357]KLxZcuDMpXiy+NEMVrhZTJ4+zRoFLaIkL7BCBKYZHopLfR2pAHq1/485SzFx2qy2[/tex][br][/br]

举一反三

内容

0
设球体占有闭区域[tex=15.286x1.286]xfDEWnOcvIE9Qgr+ufTWG0fWiuZLzHNk83YLFP58rFYIsi8vNuqmRhiZOBY5UZxFpgiP696ddsA/MCATipqEocl2vwN55IuXSfKd4S+zPzQ=[/tex]，它在内部各点的密度的大小等于该点到坐标原点的距离的平方，试求这球体的质心。
1
设球体 [tex=3.929x1.286]OgRXGBnuYUkrpNulxRW68D36NV9X5hevhTpuCfbJIg4=[/tex][tex=3.286x1.286]8UBoqWgIU0LEZK9ye4gOwmYF7i4S+RlL7M++VMzjL0E=[/tex] 上各点的密度等于该点到坐标原点的距离,求这个球体的质量.
2
设球体[img=149x26]1802e977c538e3a.png[/img]内每一点[img=56x25]1802e977cd3ad1e.png[/img]处密度的大小等于该点到坐标原点距离的平方，则该球体的质心坐标为（）。未知类型：{'options': ['', '', '', ''], 'type': 102}
3
设球体[img=149x26]1802e97ad2a8757.png[/img]内每一点[img=56x25]1802e97adaab9c3.png[/img]处密度的大小等于该点到坐标原点距离的平方，则该球体的质心坐标为（）。未知类型：{'options': ['', '', '', ''], 'type': 102}
4
设球体[img=149x26]1802e952c8cddbd.png[/img]内每一点[img=56x25]1802e952d1ba4e1.png[/img]处密度的大小等于该点到坐标原点距离的平方，则该球体的质心坐标为（）。未知类型：{'options': ['', '', '', ''], 'type': 102}