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Principal Axes

We state, without proof, that from any point P of a rigid body (of any shape), we can draw three mutually perpendicular axes about which the six products of inertia of the body vanish. These axes are called the principal axes.

Hence, if we choose body-axes at P as the principal axes, then we have,

I_xy = I_xz = I_yz ….. = 0

and therefore, (LP)_x = I_xx ω_x, (LP)_y = I_yy ω_y, (LP)_z = I_zz ω_z

where I_xx etc. are the moments of inertia about the principal axes at P.

How do we determine the principal axes? For regular bodies, lines of symmetry at any point P turn out to be the principal axes, as illustrated below for some simple systems:

(i) Sphere: In case of sphere, one of the principal axes, at any point P in the sphere, is the line joining P and the center C of sphere. The other two axes are any two mutually perpendicular lines in the plane normal to the line PC (the first axis).

(ii) Cylinder: One principal axis (PZ) at any point P in the cylinder is a line from P parallel to the axis of the cylinder. The other principal axis (say PX) is the perpendicular drawn from P on the axis of the cylinder. The third axis is obviously the line at P normal to both PX and PZ. Same geometry holds for a disc, or ring.

(iii) For a thin rod, one axis is along the length of the rod. The other two axes are any two mutually perpendicular lines in the plane normal to the length of the rod.

We shall therefore consider the motion of only regular rigid bodies like spheres, cylinders, etc. so that we can work with principal axes only.