198 



MOTION UNDER CONSTANT FORCES 



163. Frame moving with vertical acceleration. If the frame 

 of reference moves with an acceleration a vertically downwards 

 we see that before measuring accelerations relative to this frame 

 we must suppose the vertical component of force on each particle 

 of mass m to be diminished by ma. Whatever forces are acting 

 there will be amongst them the weights of the particles mg, etc 

 We can conveniently suppose the diminution ma to be taken 

 from these, so that the weight of a particle, instead of being 

 taken to be mg, will be taken to be m (g a). 



Thus the acceleration of the frame of reference may be allowec 

 for by supposing the acceleration due to gravity to be diminishec 

 from g to g a. 



For example, if an Atwood's machine is placed in an elevator, then at 

 the instant at which the elevator has an upward acceleration a, the accel 

 eration of the masses relative to the machine will be (cf . equation (49)) 



while the tension of the string will be (cf . equation (50)) 



o 



164. Effect of earth's rotation on the value of g. As we have 

 seen ( 25), a frame of reference which is fixed relatively to the 



earth's surface possesses an acceleration 

 in consequence of the rotation of the 

 earth about its axis. 



Let 00' be the earth's axis, and let 

 P be any point on the earth's surface 

 in latitude X. Regarding the earth as a 

 sphere of radius a, the point P will de- 

 scribe a circle of radius a cos X having 

 its center N on the earth's axis. If v is 

 the velocity with which P describes the 

 circle, the acceleration of P is, by 12 



a cosX 



towards the center of the circle, i.e. along PN. 



