기타 풀이/회전관성

속이 채워진 원기둥의 회전 관성

짱세디럭스 2024. 5. 7. 23:52

[Rotational Inertia of Solid Cylinder]\title{Rotational Inertia of Solid Cylinder} ρ= ⁣dm ⁣dv=MV=MAH=MπR2H(1) \rho=\frac{\dd m}{\dd v}=\frac{M}{V}=\frac{M}{AH}=\frac{M}{\pi R^2 H} \taag1 l=rθ, ⁣dl=r ⁣dθ \begin{aligned} l&=r\theta,\\ \dd l&=r \dd \theta\\ \end{aligned}  ⁣da= ⁣dl ⁣dr=(r ⁣dθ) ⁣dr=r ⁣dr ⁣dθ(2) \begin{aligned} \dd a&=\dd l\cdot\dd r \\ &=(r \dd \theta)\cdot\dd r \\ &=r \cdot \dd r\dd \theta \taag2\\ \end{aligned}  ⁣dv= ⁣da ⁣dh=(r ⁣dr ⁣dθ) ⁣dh=r ⁣dr ⁣dθ ⁣dh(3) \begin{aligned} \dd v&=\dd a \cdot \dd h\\ &=(r \dd r \dd \theta) \cdot \dd h\\ &=r \cdot \dd r \dd \theta \dd h\taag3\\ \end{aligned}  ⁣dm=ρ ⁣dv=ρ(r ⁣dr ⁣dθ ⁣dh)=ρr ⁣dr ⁣dθ ⁣dh(4) \begin{aligned} \dd m&=\rho\cdot\dd v\\ &=\rho\cdot (r \dd r \dd \theta \dd h)\\ &=\rho r\cdot\dd r \dd \theta \dd h\taag4\\ \end{aligned}  ⁣dI=r2 ⁣dm=r2(ρr ⁣dr ⁣dθ ⁣dh)=ρr3 ⁣dr ⁣dθ ⁣dh(5) \begin{aligned} \dd I&=r^2\cdot \dd m\\ &=r^2\cdot \(\rho r\dd r \dd \theta \dd h\)\\ &=\rho r^3\cdot\dd r \dd \theta \dd h\taag5\\ \end{aligned} ISolid Cylinder=V ⁣dI=0H02π0Rρr3 ⁣dr ⁣dθ ⁣dh=ρ(0Rr3 ⁣dr)(02π ⁣dθ)(0H ⁣dh)=(MπR2H)(14R4)(2π)(H)=12MR2 \begin{aligned} I_{\text{Solid Cylinder}}&=\oiiint_V \dd I\\ &=\int_0^{H}\int_0^{2\pi}\int_0^{R} \rho r^3\dd r \dd \theta \dd h\\ &=\rho\cdot\(\int_0^{R} r^3\dd r\)\cdot\(\int_0^{2\pi} \dd \theta\)\cdot\(\int_0^{H} \dd h\)\\ &=\(\frac{M}{\pi R^2 H}\)\cdot\(\frac{1}{4}R^4\)\cdot(2\pi)\cdot(H)\\ &=\frac{1}{2}MR^2\\ \end{aligned}