Question

The solid cylinder of length $80~\text{cm}$ and mass M has a radius of $20~\text{cm}$. Calculate the density of the material used if the moment of inertia of the cylinder about an axis CD parallel to AB as shown in figure is \(2.7 ~\text{kg}~ \text m^2\).

• A. $14.9\text{kg}{\text{m}}^{-3}$
• B. $7.5\times {10}^1\text{kg}{\text{m}}^{-3}$
• C. $7.5\times {10}^2\text{kg}{\text{m}}^{-3}$
• D. $1.49\times {10}^2\text{kg}{\text{m}}^{-3}$

Answer 1

The correct option is D $1.49\times {10}^2\text{kg}{\text{m}}^{-3}$

The moment of inertia about an axis passing through its center along AB is $\frac{M{r}^2}{2}$, and that at a distance of $L/2$ is $M{d}^2=M{\left(\frac{L}{2}\right)}^2$.

Applying parallel axis theorem we have,

$I=I_c+M{d}^2$

i.e. $I=\frac{M{r}^2}{2}+\frac{M{L}^2}{4}$

$2.7=M\frac{{0.2}^2}{2}+M\frac{{0.8}^2}{4}$

$2.7=M[\frac{2}{100}+\frac{16}{100}]$

So, $M=\frac{270}{18}=15\text{kg}$

The required density of the cylinder is,

$\rho =\frac{M}{\pi r^{2}L}=\frac{15}{3.14\times {0.2}^2\times 0.8}=0.149\times {10}^3\text{kg}{\text{m}}^{-3}$