Show that the moments of inertia of any body satisfy the triangle inequalities $$ I_{3} \leq I_{2}+I_{1} \quad I_{2} \leq I_{1}+I_{3} \text { and } I_{1} \leq I_{2}+I_{3} \text {. } $$ and that equality holds only for a planar body.

The concept of the inertia tensor is integral to understanding how an object's mass distribution affects its rotational properties. Imagine a rigid body as an assortment of masses scattered in space, each particle contributing to how difficult it is to spin the object around any given axis.

The inertia tensor is a mathematical concept that takes all these contributions and organizes them into a matrix that links the angular velocity of an object to its angular momentum. It is crucial in physics because it offers a comprehensive description of an object's inertia in all directions. The tensor is expressed as a 3x3 matrix, embodying not only the individual moments of inertia along three perpendicular axes but also the product of inertia which accounts for mass distribution in mixed planes.

Understanding and calculating the inertia tensor enable physicists and engineers to predict the rotational motion of bodies more accurately, crucial for applications ranging from mechanical engineering to aerospace dynamics.

The principal moments of inertia are special values that are derived from the inertia tensor. They represent the moments of inertia along three mutually perpendicular axes called principal axes. These axes are unique to the body's geometry and mass distribution and align with the directions in which the mass is most and least spread out.

To find the principal moments of inertia, one typically computes the eigenvalues of the inertia tensor. These eigenvalues, denoted as I₁, I₂, and I₃, have a profound physical meaning: they are the moments of inertia for rotations around the principal axes, where the products of inertia are zero. These values are fundamental since they provide the simplest picture of the object's rotational dynamics, free from the complications of off-diagonal terms (products of inertia) in the tensor matrix.

The step-by-step solution provided describes how these principal moments are not arbitrary but satisfy an important set of inequalities, akin to the triangle inequalities found in geometry.

Triangle inequalities are most commonly known from geometry, where they state that the length of any side of a triangle must be less than or equal to the sum of the lengths of the other two sides. A similar principle applies in the context of physics, particularly with moments of inertia of three-dimensional objects.

These inequalities become significant when ensuring the physical plausibility of the rotational dynamics of objects. The solution outlined step-by-step in the textbook demonstrates how, for moments of inertia, each value I₁, I₂, and I₃ must adhere to the conditions resembling those of the sides of a triangle. This ensures that the calculated moments of inertia correspond to a physically realizable body.

These constraints are not just mathematical curiosities; they are grounded in the physical reality of how mass can be distributed in space. Mass cannot be distributed in a way that violates these inequalities, just like in the real world, you cannot construct a triangle with sides that do not adhere to this rule.

To further grasp the concept of moments of inertia, it's enlightening to explore the special case of a planar body. A planar body is one that has all of its mass distributed along a single plane. Common examples are flat plates, thin sheets, or any object whose thickness can be disregarded in comparison to its other dimensions.

In the case of a planar body undergoing rotation, one moment of inertia (typically I₃, if the object lies in the xy-plane) is significantly different from the other two (I₁ and I₂). This is because there's no mass outside the plane to influence rotations about the axis perpendicular to it. The exercise solutions highlight that for such a body, the moments of inertia satisfy the triangle inequalities in a state of equality: I₃ equals the sum of I₁ and I₂.

This unique characteristic allows for simplified calculations and analysis of the body's rotational behaviour, making it an excellent starting point for beginners to the study of rotational dynamics before progressing to the complexities of three-dimensional bodies.