Question

Find dimension of moment of inertia. Is moment of inertia a vector or a scalar?

Answer 1

Hint
To find the dimension of moment of inertia, we need to use the formula for the moment of inertia to find its units. From the dimensions of the various terms in the formula of moment of inertia, we get the dimension of moment of inertia.

Formula Used: In this solution we will be using the following formula,
 $ I = M{R^2} $
where $ I $ is the moment of inertia
 $ M $ in the mass of the body
 $ R $ is the radius of gyration.

Complete step by step answer
The moment of inertia of a body is its ability to resist angular acceleration. It determines the amount of torque that a body needs for a desired angular acceleration about a rotational axis. The moment of inertia of a body is given by the sum of the moments of inertia of each particle, and for each particle the moment of inertia is the product of the mass and square of the distance of the particle from the rotational axis.
Therefore we can write the moment of inertia as,
 $ I = M{R^2} $
Now mass has a unit of length. So we can write the dimension of mass as,
 $ M = \left[ {{M^1}{L^0}{T^0}} \right] $
The radius of gyration is the unit length. So we can write the radius as,
 $ R = \left[ {{M^0}{L^1}{T^0}} \right] $
So the moment of inertia has a dimension of,
 $ I = \left[ {{M^1}{L^0}{T^0}} \right]{\left[ {{M^0}{L^1}{T^0}} \right]^2} $
We can write this as,
 $ I = \left[ {{M^1}{L^2}{T^0}} \right] $
So the dimension of moment of inertia can be calculated as $ \left[ {{M^1}{L^2}{T^0}} \right] $ .
The moment of inertia is the rotational equivalent of mass in the translational motion.
Now since mass is a scalar quantity, so the moment of inertia is also a scalar quantity.

Note
The dimensional formula is an equation which gives the relation between the fundamental units and the derived units in the terms of dimensions. In mechanics we take the length, mass and time as the three fundamental dimensions and they are represented by $ M $ , $ L $ and $ T $ .