Question

What is the moment of inertia of a rod with a mass of $6\,kg$ and length of $9\,m$ that is spinning around its centre?

Answer 1

In this question, it is given that the rod is spinning about the centre. So, the axis of rotation is passing through the centre of the rod. Also, for a rod the geometrical centre is the centre of mass of the rod. So, the axis passes through the centre of mass. We will use the formula of moment of inertia of a rod when the axis passes through the centre of mass and substitute the values in it to get the answer.

Complete step by step answer:
Moment of inertia in rotational dynamics is the analog of the mass in linear motion. For a point mass, it is the product of its mass and square of the distance from the axis of rotation.Moment of inertia tells the relative distribution of mass in a rigid body. For a rod, the moment of inertia can be defined for two general cases:
-When the axis is passing through the centre of the rod
-When the axis is passing through the end of the rod

In this question, it is given that the rod is spinning about the centre. So, the axis of rotation is passing through the centre of the rod.This is also the centre of mass of the rod.The moment of inertia of a rod when the axis of rotation passes through the centre of mass is given by,
$I = \dfrac{{m{l^2}}}{{12}}$
where $m$ is the mass of the rod and $l$ is the length of the rod.
We are given that $m = 6\,kg$ and $l = 9\,m$
Substituting the values we have,
$I = \dfrac{{6 \times {9^2}}}{{12}}$
$\therefore I = \dfrac{{45}}{2} = 22.5$

Therefore, the moment of inertia of a rod is $22.5$.

Note: The moment of inertia is specified for a particular axis of rotation. If the axis of rotation passes through the end of the rod, the moment of inertia is $I = \dfrac{{m{l^2}}}{3}$ which is different from the moment of inertia when the axis passes through the centre. The moment of inertia would also vary if the rod has non uniform mass density. In this question, nothing was specified for the mass density of the rod and so we assumed it to be a constant.