Question

Calculate the moment of Inertia of a solid cylinder of mass '$M$' and radius $R$ about its Axis(as shown)?

• A. $M{R}^2$
• B. $\frac{M{R}^2}{4}$
• C. $\frac{M{R}^2}{2}$
• D. $\frac{M{R}^2}{3}$

Answer 1

The correct option is C

$\frac{M{R}^2}{2}$

Draw two cylindrical surfaces of radii $x$ and $x+dx${as shown in figure}. Consider the part of the cylinder

in between the two surfaces given length of the cylinder = $h$

Mass per unit volume of the cylinder = $\frac{M}{\pi R^{2}h}$

Then the mass of hollow cylinder cinsidered = $dm$

= $\frac{M}{\pi R^{2}h}\times 2\pi xdx\times h$

= $\frac{2M}{R^2}\times xdx$ ..........(i)

As its radius is $x$, its moment of Inertia about the given axis = $dm.x^2$

$\Rightarrow I$ = $\underset{0}{\overset{R}{\int }}{x}^{2}dm$

From (1)

$I$ = $\underset{0}{\overset{R}{\int }}\frac{2M}{R^2}{x}^{3}dx$ = $\frac{M{R}^2}{2}$

Note: It does not depends upon the length (i.e. $h$) of the cylinder.