Question

怎么证明圆球的转动惯量是2/5mr²

Answer 1

体积密度 σ=m/(4πr^3/3)=3m/(4πr^3)

薄圆片体积 (πy^2)dx

薄圆片对x轴转动惯量 σ( 锋让念πy^2)dx.y^2/2=(σ πy^4)/2dx

y=√(r^2-x^2)

球对x轴转动惯量 Jx

 Jx =2∫(σ πy^4)/2dx=(σπ(√(r^2-x^2))^4)dx=σπ(r^5+r^5/5-2r^5/3)=σπ(8r^5/15)

 银困滑斗     (0-->r)

 将σ=3m/(4πr^3) 代入

 Jx=π3m/(4πr^3)(8r^5/15)=2mr^2/5

 *Jx=Jy=Jz