H7-] 



PLANE MOTION. 



75 



As P 1 P 2 = A 1 A 2 = 2a, we may regard A^A^ as representing the 

 stroke. The position of the piston head P at the time when the 

 crank pin is at Q will then be found as the intersection N oi a 

 circle of radius / described about P with the diameter A 



Fig. 35. 



of the crank circle ; in other words, N represents the position of 

 the piston corresponding to the angle A 1 QQ = in the forward 

 stroke and to the angle A 1 OQ f = 27r in the 'return stroke. 



146. The crank may generally be assumed to turn uniformly, 

 making n revolutions per second. The linear velocity of the 

 crank pin Q is therefore u = 2Tra n = Trns. 



For the piston head P t or for the point N, we must distin- 

 guish between its mean, or average, velocity V, and its variable 

 instantaneous velocity v at any particular moment. For each 

 revolution of the crank the. piston head completes a double stroke 

 so that its mean speed is V2ns. Hence we have 



u 



_ TT 

 2ns~ 2 



147. The instantaneous velocity v of the piston can be found 

 graphically by considering the motion of the connecting rod 

 PQ. The velocity u of the end Q is known, both in magnitude 

 and direction ; the velocity v of the other end is known in direc- 

 tion only. Now considering that the length of the rod PQ is 

 invariable and hence the components of u and v along PQ must 



