MOVING FRAME OF REFERENCE 197 



An acceleration of this smallness could easily be measured ; for instance, 

 the heavier mass would only descend eight feet in ten seconds. In prac- 

 tice, the difficulty arises that if the difference of the weights is made too 

 small, the forces acting on the pulley are so evenly balanced that their 

 difference is not sufficient to overcome the friction of the bearings, etc. 



MOTION REFERRED TO A MOVING FRAME OF REFERENCE 



162. It has already been seen (25) that the second law of 

 motion remains true when the motion is measured relative to a 

 frame of reference which is not at rest but is moving with a uni- 

 form velocity. It is easy to find how the statement of this law 

 must be modified when the frame of reference moves with a known 

 acceleration. 



Let a be the acceleration of the frame of reference, let / be the 

 component of the acceleration of a moving particle in the direction 

 of the acceleration a, and let P be the component in this direction 

 of the force acting on the particle. By the second law of motion 



P = mf, (51) 



where /' is the component acceleration referred to a frame of refer- 

 ence at rest. The acceleration/' may, however, be regarded as com- 

 pounded of the acceleration / of the particle relative to the moving 

 frame of reference, together with the acceleration a of this frame 

 relative to one at rest. Since these accelerations are all in the 

 same direction, we have f f =f-}-a,so that equation (51) becomes 



P = m(f+a). 



We can also write this in the form 



P ma = mf, (52) 



showing that the motion is the same as if the frame were at rest, 

 provided we imagine the force P diminished ~by an amount ma. 



This result can easily be interpreted physically. Of the force P, 

 a part equal to ma is used up in causing the particle to keep pace 

 with the moving frame of reference. It is only the remaining part, 

 P ma, which is available for producing accelerations relative to 

 the moving frame. 



