3I4-] BODY WITH FIXED POINT. 



IV. Rigid Body with a Fixed Point. 



311. A rigid body with a fixed point has three degrees of 

 freedom. Any one of its points, with the exception of the 

 fixed point O, is constrained to the surface of a sphere and has 

 therefore two degrees of freedom ; and the body itself can turn 

 about the line joining this point to O. The motion consists, at 

 any instant, of an infinitesimal rotation about an axis passing 

 through O (see Part L, Arts. 32-35). Both the angular velocity 

 and the direction of the instantaneous axis vary in the course 

 of time. 



312. We begin with the study of the instantaneous motion of 

 the body, which may be regarded as due to the action of a sys- 

 tem of impulses on the body at rest. This will lead to the solu- 

 tion of the converse problem, viz. to determine the initial motion 

 produced by a given system of impulses. 



I. INITIAL MOTION DUE TO IMPULSES. 



313. The body rotates at the time /with angular velocity o> 

 about the instantaneous axis / which passes through the fixed 

 point O. It is required to determine a system of impulses that 

 would produce this motion if acting on the body at rest. 



For O as origin, let R be the resultant and H the result- 

 ing couple of these impulses. If the impulsive reaction A of 

 the fixed point O be combined with them, the body can be 

 regarded as free, and its instantaneous motion is determined by 

 the equations (19) and (20), Art. 238. It is only necessary, in 

 the equations (19), to add to the components R XJ R y , R z of R 

 those of A, while the right-hand members of (20) are not 

 affected by A, since its moment is zero for every axis through O. 



314. It remains to form the sums in the left-hand members 

 of (19) and (20) for our case ; i.e. to reduce the system of 

 momenta mx, my, mz of the particles to its resultant and result- 

 ant couple for a fixed rectangular system of axes through O. 



