298 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



along and perpendicular to AB, and the velocity of P relative to O has 

 components 



and 



in the same directions. 



Hence the moment of momentum in the motion relative to G is 



or ?a 2 



also twice the kinetic energy in the motion relative to G is 



j) COS*]. 



Now the centre of inertia moves with uniform velocity in a straight line ; 

 and thus the kinetic energy of the whole mass placed at the centre of inertia 

 and moving with it is constant, and the moment about any fixed axis of the 

 momentum of the whole mass placed at the centre of inertia and moving 

 with it is alsa constant. Also the kinetic energy of the system and its 

 moment of momentum about any fixed axis are constants. Hence the 

 moment of momentum in the motion relative to G and the kinetic energy 

 in the same relative motion are constants. 



Let V be the velocity with which the particle was initially projected at 

 right angles to the thread; then the initial values of the moment of 

 momentum and kinetic energy in the motion relative to G are 



(a + 1) Vmpl(m +p}, and \ V 2 mp/(m +p). 

 Hence throughout the motion we have the equations 



*263. Kinematical Note. It is sometimes convenient in calculating 

 the velocities of points in a connected system to use the coordinates of a 

 point referred to axes which do not retain the same direction. Thus in the 

 problem of Article 262 we might have obtained the velocity of P relative to 

 M by taking as axes lines through M along and perpendicular to AB. When 

 we wish to calculate the velocity of a point in this way we have to attend 

 to the fact that the component velocities parallel to the moving axes are 

 not the differential coefficients (with respect to the time) of the coordinates 

 referred to the same axes. 



Consider the motion of a particle P whose coordinates at time t are a/, y 

 referred to rectangular axes rotating in their own plane about the origin ; 

 let be the angle which the axis of x makes with a fixed axis of x in the 

 plane at time t, and #, y the coordinates of the particle referred to fixed 



