A symmetric object is one that has the same shape, size, and mass distribution on both sides. This means that if the object is divided in half along any axis, the two halves will be identical.
The center of mass is the point at which the entire mass of an object can be considered to be concentrated. When the principle axis goes through the center of mass, it ensures that the object will rotate smoothly without any wobbling or tipping.
A common way to prove this is to use the parallel axis theorem, which states that the moment of inertia of an object can be calculated by adding the moment of inertia about an axis through the center of mass with the product of the object's mass and the square of the distance between the two axes. If the moment of inertia is the same about all axes passing through the center of mass, then the principle axis must go through the center of mass.
The principle axis is an important concept in rotational dynamics. It is the axis about which an object will rotate with the least resistance or energy expenditure. For a symmetric object, the principle axis will always go through the center of mass, making it a useful tool for analyzing the motion and stability of such objects.
In theory, yes, but in practice, this is highly unlikely. For a symmetric object, the center of mass is often a point of symmetry, and the principle axis will always go through this point. However, if the object is not perfectly symmetric or has an irregular mass distribution, it is possible for the principle axis to not go through the center of mass. This would result in a more complex rotational motion for the object.