Moment Of Inertia Of The Solid Cylinder - Practice Questions & MCQ

Let I= Moment of inertia of the CYLINDER about an axis through its centre

To calculate I

Consider a cylinder of mass M, radius R and length L. 

mass per unit volume of the cylinder   $\rho=\frac{M}{V}=\frac{M}{\pi R^2 L}$

Imagine that the cylinder is made of a large number of coaxial cylindrical shells

Take small elemental cylindrical shell of mass dm having internal radius x and external radius (x + dx).

So for that elemental cylindrical shell $d V=(2 \pi x d x) L$

And

$
\begin{aligned}
& \text { And } d m=\rho d V=\frac{M}{\pi R^2 L}(2 \pi x d x) L \\
& \Rightarrow d I=x^2 d m
\end{aligned}
$

Now integrate this dl between the limits $x=0$ to $x=R$

$
\begin{aligned}
& I=\int d I=\int x^2 * \rho d v \\
& =\int_0^R \frac{M}{\pi R^2 L}\left(2 \pi * L x^3 d x\right) \\
& =\frac{2 M}{R^2} \int_0^R x^3 d x=\frac{M R^2}{2}
\end{aligned}
$