KINETICS OF A RIGID BODY. 



[300. 



always, and are the only, permanent axes of rotation. This can 

 be regarded as the dynamical definition of principal axes. 



300. It appears from the equations (5) that the position of 

 the axis of rotation will remain the same if, in addition to the 

 absence of external forces, the conditions 



^=0, j/ = o (8) 



be fulfilled ; for in this case the components of the force (5) al 

 vanish. If, moreover,, the axis of rotation be a principal axis 

 the rotation will continue to take place about the same line even 

 when the body has no fixed point. 



The conditions (8) mean that the centroid lies on the axis ol 

 z\ and it is known (Art. 264) that a centroidal principal axis is 

 a principal axis at every one of its points. The axis of z must 

 therefore be a principal axis of the body, i.e. a principal axis at 

 the centroid. We have, therefore, the proposition : If a fret 

 rigid body, not acted upon by any forces, begin to rotate about one 

 of its centroidal principal axes, it will contimte to rotate uniformly 

 about the same line. 



301. A rigid body with a fixed horizontal axis is called a 

 compound pendulum if the only external force acting is the 

 weight of the body. 



The plane through axis and centroid will make, with the 

 vertical plane (downwards) through the axis, an angle 0, which 

 we may take as angle of rotation, so that 

 a) = d6/dt (Fig. 35). The weights of the par- 

 ticles, being all parallel and proportional tc 

 their masses, have a single resultant Mg pass 

 ing through the centroid G. Hence, if h be 

 the perpendicular distance OG of the centroic 

 from the axis, the moment of the externa 

 forces is H= Mgh sin 6 ; and if the radius oi 

 inertia of the body for the centroidal axis 

 parallel to the axis of rotation be q, the moment 

 of inertia for the latter axis is I= 



Ma 



Fig. 35. 



