Question

Two discs of the same material and thickness have radii $0.2m$ and $0.6m$. Their moments of inertia about then axes will be in the ratio of?

• A. $1:3$
• B. $1:9$
• C. $1:27$
• D. $1:81$

Answer 1

The correct option is D

$1:81$

The explanation for the correct option:

Option D

Step 1: Given that:

Two-disc having the same thickness ($t$) and made up of the same material($\rho$ density) having radii,

$$\begin{gathered} r_1=0.2m \\ {r}_2=0.6m \end{gathered}$$

Step 2: Formula used:

The momentum of inertia($I$), of a disk about its central axis, is given by

$I=\frac{1}{2}m{r}^2....\left(1\right)$

where $m$is the mass of the disc

$r$ is the radius of the disc

Now the mass of a disc is given by,

$m=\pi r^{2}t\times \rho .....\left(2\right)$

So, the ratio is given by,

$\begin{array}{rcl}\frac{I_1}{I_2}& =& \frac{{\displaystyle \frac{1}{2}}{m}_1{r}_1^2}{{\displaystyle \frac{1}{2}}{m}_2{r}_2^2}\\ & =& \frac{\pi r_1^{2}t\times \rho \times r_1^2}{\pi r_2^{2}t\times \rho \times r_2^2}\\ & =& \frac{r_1^4}{r_2^4}\\ & =& {\left(\frac{0.2}{0.6}\right)}^4=\frac{1}{{\left(3\right)}^4}\\ & =& \frac{1}{81}\end{array}$

Hence the ratio between the moments of inertia is $1:81$.