求质量均匀分布的物体:x^2/a^2+y^2/b^2+z^2/c^2=0,y>=0,z>=0的形心求详解过程,谢谢!

Question

求质量均匀分布的物体:x^2/a^2+y^2/b^2+z^2/c^2=0,y>=0,z>=0的形心
求详解过程,谢谢!

Answer 1

设A=∫∫∫dV=πabc/3
B1=∫∫∫xdV=∫∫∫xdxdydz=∫∫∫arsinφcosθ *abcr^2sinφdrdθdφ
=a^2bc∫(0->1)dr ∫(0->π/2)dθ ∫(0->π/2) r^3 (sinφ)^2 cosθdrdθdφ
=a^2bcπ/16
同理B2=∫∫∫ydV=∫∫∫brsinφsinθ *abcr^2sinφdrdθdφ=ab^2cπ/16
B3=∫∫∫zdV=∫∫∫crcosφ *abcr^2sinφdrdθdφ=abc^2π/16
所以质心x0=B1/A=3a/16,y0=B2/A=3b/16,z0=B3/A=3c/16
所以质心(3a/16, 3a/16, 3a/16)