34 FORCE AND THE LAWS OF MOTION 



referred to axes fixed in space or to the moving' axes ; for the 

 acceleration referred to the moving axes is obtained by compound- 

 ing the acceleration referred to axes fixed in space with that of the 

 moving axes, and this latter acceleration is nil. 



Thus it appears that the laws of motion retain exactly the same 

 form when the motion is referred to axes which move in space, 

 provided that these axes move with no acceleration. 



This condition of no acceleration is not satisfied by a set of 

 axes fixed in the earth's surface. A point on the earth's surface 

 describes, on account of the earth's rotation, a circle about the 

 earth's axis. If a is the radius of this circle, and v the velocity 

 with which it is described, the point will have, by 12, an accelera- 



v 2 



tion towards the earth's axis of rotation. Thus a set of axes 

 a 



fixed in the earth's surface will have an acceleration of this 

 amount, and this has to be borne in mind in applying the laws of 

 motion. At a point on the equator v = 46,510 centimeters per 

 second and a = 637 X 10 6 centimeters, so that the acceleration is 



v 2 



= 3.4 centimeters per second per second. A body dropped at the 



a 



equator will appear to have an acceleration of 978.1 centimeters per 

 second per second, if the motion is referred to axes fixed in the 

 earth ; but will have a true acceleration of amount 



978.1 + 3.4 = 981.5, 

 if the acceleration is referred to axes fixed in space. 



This explains part of the reason why the force of gravity appears 

 to vary from point to point at the earth's surface. The weight of 

 a mass of one kilogramme will produce a certain extension of the 

 spring of a spring balance at the North Pole. If taken to the 

 equator, part of the weight goes towards producing the acceleration 

 of tnVmass towards the earth's center, and it is only the remain- 

 der which extends the spring of the balance. The first part is the 

 weight of about 31 grammes ; the remainder is the weight of about 

 996J- grammes. Thus we may say that, owing to the acceleration 

 of the earth's surface towards its center, a mass of a kilogramme 



