Question

A rod is lying along x-axis with its left extreme at origin as shown in figure. The density of the rod increases by the rate, $\frac{dm}{dx}=5x\frac{kg}{m}$. The length of the rod is 2m. What will be its rotational inertia about Z axis?

• A. $\frac{10}{3}kg-m^2$
• B. $\frac{20}{3}kg-m^2$
• C. $10kg-m^2$
• D. $20kg-m^2$

Answer 1

The correct option is D $20kg-m^2$

Let's take a small length element 'dr' at a distance 'r' as shown in figure.

Now, dm = 5r dr
$I={\int }_0^2\left(r^2\right)\left(5rdr\right)$
$=\left(5\right){\left[\begin{array}{c}\frac{r^4}{4}\end{array}\right]}_0^2=\frac{5}{4}\left\{\begin{array}{c}16-0\end{array}\right\}$
$=20kg-m^2$