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NOTES ON EELATIVE MOTION. 



BY JAMES LOUDON, 

 Uniifersity College, Toronto. 



1 . Motion of a point in a plane. 



At time t let the moving axes be 0^, Orj, and P a point (? , ij) in 

 their plane. At time f -\- dt let these axes coincide with 0^', Or}', 

 and P with P' ; then the ^ and t; components of the displacement 

 FP' ai'e — wYjdt, (u^dt, respectively, if w is the rate at which the axes 

 turn round 0?. Let a moving point he at P at time t, and at Q at 

 time t -(- ^i, the co-ordinates of Q referred to Of, Orj' being ? -|- |5f, 

 ly -j- )j5i ; then the absolute velocity of the moving point is ultimately 



PQ fPP' P'Q\ 



-^^{ -Tr> -^ ), the f and ij components of which are ? — wr), rj -j- 



a»?, respectively. 



Putting ^ — 107] = u= OA, and yj -f- o? = -y ^ 05, the component 

 velocities at time t -{- dt become u + iidt =^ OA' along 0^', and v -j- 

 «<Jf = 05' along 0-q'. Hence the absolute acceleration ultimately = 



(AA' BB'\ 

 -r-> -jr], the components of which are 



u — vu) ^ ^ — 2 cDfi — 7]d> — (tf^f along 0^, 

 V -}- uw =z7i-^2o)^-\-^d) — (o'^Tfj along Otj. 



2. Motion of a rigid body round a fixed axis OC, the axes Of, Otj 

 being fijced in the body. 



At time t the whole momentum is — 3fui7] = OA along 0$, and 

 Mm? = OB along Orj, where f , yj are co-ordinates of the centre of 

 inertia. At time t -{- 86 the momentum is — Mtj (tw -\- (hdt) = OA' 

 along Of, and M? {m -\- &dt) = OB' along 0);'. The changes of 



momentum, per unit time are, therefore, ultimately —t-^ -^> whose 



components are 



— Mrjd) — McD^$ along 0$, 

 M?(a — Mio^-q along Orj. 

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