Question

How is the moment of inertia a tensor?

Answer 1

A tensor is an algebraic object in mathematics that represents a (multilinear) connection between sets of algebraic objects in a vector space. Tensors may map between vectors and scalars, as well as between other tensors. Scalars and vectors (the simplest tensors), dual vectors, multilinear mappings across vector spaces, and even certain operations like the dot product are all examples of tensors.

Complete answer:
A rigid body's moment of inertia is a number that defines the torque required to achieve a desired angular acceleration along a rotating axis, similar to how mass influences the force required to achieve a desired acceleration. It relies on the mass distribution of the body and the axis selected, with greater moments necessitating more torque to affect the rate of rotation. It is an extended (additive) property: the moment of inertia for a point mass is simply the mass times the square of the perpendicular distance to the rotation axis. A rigid composite system's moment of inertia is equal to the sum of the moments of inertia of its component subsystems.
Torque must be given to a body that is free to spin around an axis in order to modify its angular momentum. The moment of inertia of the body is related to the amount of torque required to generate any given angular acceleration (rate of change in angular velocity). In SI measurements, the moment of inertia is measured in kilograms metre squared ($kg{m^2}$). In rotational kinematics, the moment of inertia serves the same purpose as mass (inertia) in linear kinetics: both describe a body's resistance to changes in motion. The moment of inertia is determined by how mass is distributed around a rotating axis, and it varies depending on the axis used.
Because it behaves as a scalar and a vector, the moment of inertia is a tensor. This is because the moment of inertia is affected by the direction and the mass distribution of the particles in the object at different moments.

Note:
It's typical in applications to look at scenarios where a distinct tensor might arise at each point of an item; for example, the tension within an object can change from one position to the next. The idea of a tensor field emerges as a result of this. Tensor fields are so common in some sectors that they are simply referred to as "tensors."