Prove Steiner's theorem: The moments of incrtia of any rigid body relative to two parallel axes, one of which passes through the center of mass, are related by the equation $$ I-I_{0}+m r^{2}, $$ where \(m\) is the mass of the body, \(r\) is the distance between the axes, and \(I_{0}\) is the moment of inertia relative to the axis passing through the center of mass. Thus the moment of inertia relative to an axis passing through the center of mass is less than the moment of inertia relative to any parallel axis.

The moment of inertia, often symbolized as 'I', is a crucial concept in rigid body dynamics, expressing a body's tendency to resist angular acceleration. This resistance is analogous to mass in linear motion, but for rotational movements instead.

Mathematically, it's the sum of all the mass particles within an object, each multiplied by the square of its distance from the rotation axis, represented as:
\[\begin{equation}I = \int r^2 dm,\end{equation}\]where 'r' is the distance from the axis, and 'dm' represents an infinitesimal element of mass.

Understanding moment of inertia is pivotal when predicting the rotational motion of a rigid body when subjected to torque. The higher the moment of inertia, the less easily the body will rotate about the axis.

Rigid body dynamics is the study of the motion of rigid bodies that do not deform when subjected to forces or torques. This field within classical mechanics assumes that the distances between particles in a body remain constant, which simplistically means the body does not stretch, compress, or change shape.

Dynamics concerns itself primarily with the relationships involving forces, mass, and motion. Within this domain, rigid bodies may experience translations (linear motion), rotations (motion about an axis), or a combination called planar motion.

The equations of motion for a rigid body are more complex than for a particle because they must account for both translational and rotational movement. Equations involving the moment of inertia are integral to solving rigid body dynamics problems, as they allow for the calculation of rotational kinematics and dynamics.

The parallel axis theorem, also known as Steiner's theorem, is an essential principle in classical mechanics. It provides a way to calculate the moment of inertia of a body about any axis parallel to an axis through its center of mass, which greatly simplifies solving many physics problems.

The theorem states that the moment of inertia about a parallel axis can be obtained from the moment of inertia about a center of mass axis (denoted as \(I_0\)) with the addition of the product of the mass of the body (\(m\)) and the square of the distance between the axes (\(r\)) as:\[\begin{equation}I = I_0 + m r^2,\end{equation}\]This effectively means you can shift the reference point of the moment of inertia from the center of mass to another parallel line by taking into account the mass and the perpendicular distance between the axes.

The parallel axis theorem is highly useful in engineering and physics when an object's rotational inertia about its own center of mass is already known, but the inertia about another axis is required, such as in designing flywheels, axles, or any rotating machinery.