Using appropriate labelled diagram, obtain the kinetic energy of a free symmetrical top of mass M.

To determine the kinetic energy of a free symmetrical top of mass \( M \), we need to analyze the motion of the top, which includes both its rotational and translational kinetic energy components. A free symmetrical top, like a spinning top, typically rotates about its own axis while also potentially undergoing precession or nutation.

### 1. Coordinate System and Body Setup

We begin by considering the top as a rigid body. The top has a mass \( M \) and a moment of inertia tensor \( I \). For a symmetrical top, the principal axes of inertia are the symmetry axis (the axis around which the top spins) and two perpendicular axes in the plane of symmetry, each with the same moment of inertia. These are denoted as \( I_1 = I_2 \) and \( I_3 \) where:

- \( I_1 = I_2 \) (moments of inertia about the perpendicular axes in the plane of symmetry),
- \( I_3 \) (moment of inertia about the symmetry _________ __________ ___ ____ _________ ___.
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