1. Tardigrade
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  3. Physics
  4. A thin circular plate of mass M and radius R has its density varying as textitρ(. r .)( text=ρ) text0 textitr with ρ 0 as constant and r is the distance from its centre. The moment of Inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is I=aMR2 . The value of the coefficient a is:

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Q. A thin circular plate of mass $M$ and radius $R$ has its density varying as $\textit{ρ}\left(\right. r \left.\right)\left(\text{=ρ}\right)_{\text{0}}\textit{r}$ with $\rho _{0}$ as constant and $r$ is the distance from its centre. The moment of Inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is $I=aMR^{2}$ . The value of the coefficient $a$ is:

NTA AbhyasNTA Abhyas 2022

Solution:

$dm=\left(2 \pi r d r\right)\times \rho =2\pi \left(\rho \right)_{0}r^{2}dr$
$M=2\pi \rho _{0}\int \limits_{0}^{R} r^{2} d r=\frac{2}{3}\pi \rho _{0}R^{3}$
$I _{ CM }=\rho( dm ) r ^{2}=2 \pi \rho_{0} \int r ^{4} dr =\frac{2}{5} \pi \rho_{0} R ^{5}$
$I_{c m}=\frac{3}{5}MR^{2}$
$I=I_{c m}+MR^{2}=\frac{8}{5}MR^{2}$