Question

show that the moment of inertia about an axis passing through diameter of ring 1/2 mR 2​

Answer 1

So we've a uniform circular ring of radius $\sf{R}$ and mass $\sf{M}$ and we need to find its moment of inertia about an axis passing through one of its diameters.

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Consider an elemental length $\sf{dL}$ of mass $\sf{dM}$ which subtends an angle $\sf{d\theta}$ at the center and is inclined at an angle $\theta$ with the horizontal.

So the length $\sf{dL}$ will be given by,

$\sf{\longrightarrow dL=R\,d\theta}$

Since the ring is of uniform, the linear density of the whole ring and that of elemental length should be same.

$\sf{\longrightarrow \dfrac{dM}{dL}=\dfrac{M}{2\pi R}}$

$\sf{\longrightarrow \dfrac{dM}{R\,d\theta}=\dfrac{M}{2\pi R}}$

$\sf{\longrightarrow \dfrac{dM}{d\theta}=\dfrac{M}{2\pi}}$

$\sf{\longrightarrow dM=\dfrac{M}{2\pi}\,d\theta}$

As we see in the figure, this elemental length is at a perpendicular distance of $\sf{R\cos\theta}$ from the axis.

So its moment of inertia about that axis is,

$\sf{\longrightarrow dI=dM(R\cos\theta)^2}$

$\sf{\longrightarrow dI=\dfrac{M}{2\pi}\,R^2\cos^2\theta\,d\theta}$

And so the moment of inertia of the whole ring will be,

$\displaystyle\sf{\longrightarrow I=\int\limits_0^{2\pi}\dfrac{M}{2\pi}\,R^2\cos^2\theta\,d\theta}$

$\displaystyle\sf{\longrightarrow I=\dfrac{MR^2}{2\pi}\int\limits_0^{2\pi}\cos^2\theta\,d\theta}$

$\displaystyle\sf{\longrightarrow I=\dfrac{MR^2}{4\pi}\int\limits_0^{2\pi}(1+\cos(2\theta))\,d\theta\quad\left[\because\,1+\cos(2x)=2\cos^2x\right]}$

$\displaystyle\sf{\longrightarrow I=\dfrac{MR^2}{4\pi}\left[\theta+\dfrac{1}{2}\sin(2\theta)\right]_0^{2\pi}}$

$\displaystyle\sf{\longrightarrow I=\dfrac{MR^2}{4\pi}\left(2\pi-0+\dfrac{1}{2}\big(\sin(4\pi)-\sin0\big)\right)}$

$\displaystyle\sf{\longrightarrow I=\dfrac{MR^2}{4\pi}\cdot2\pi}$

$\displaystyle\sf{\longrightarrow\underline{\underline{I=\dfrac{1}{2}\,MR^2}}}$

Hence the moment of inertia of the ring about an axis passing through one of its diameters is $\sf{\dfrac{1}{2}\,MR^2.}$