24 I. KINEMATICS OF A POINT. LAWS OF MOTION. 



where r is the radius of the circular path. The resultant is directed 

 toward the center, and measures the effect of the tension of the string 

 on the motion. If we repeat the experiment with another body for 

 which the corresponding quantities are denoted by accents, making 

 use of the same counterbalancing weight, the tensions of the string 

 in the two cases are obviously equal and consequently we have, 



mv* m' 



r r' 



Measuring the velocities and radii therefore enables us to compare 

 the masses. 



The vector denned by the product of the scalar quantity mass 

 by the vector quantity acceleration, whose components are 



,n\ -cr d*x -r-r d*y r? d^z 



41) x = m f> Y = m -> Z = m 



is called the force acting upon the body, and is the vis impressa of 

 the second law. The second and third laws taken together accord- 

 ingly give us a complete definition and mode of measurement of force. 

 The introduction of the new term is justified by the third law. For 

 we find that force is capable of representing the dual nature of the 

 interaction between two bodies, while the acceleration is not, there 

 being two different accelerations for the two different bodies. 



The two sided nature of the action between two bodies is often 

 expressed by calling it a stress. 



The equations 41) are called the differential equations of motion 

 of the body. This statement needs some explanation. The introduc- 

 tion of the term force has given us no explanation of the cause 

 of motion, for whereas the second law tells us that the change of 

 motion is proportional to the force applied, and we are accustomed 

 to say that the force is the cause of the change, no additional 

 knowledge of the motion is given us by this statement. When we 

 say that a body moves because we push it, all we mean is that the 

 motion and the push exist simultaneously. Were we accustomed to 

 a different point of view, we might be as much struck with the fact 

 that the body pushes back when it moves as that we push it. This 

 is what the third law calls to our attention. 



It is undoubtedly true that our fundamental notions of dynamics 

 are derived through what may be called the muscular sense, which 

 is affected when we make ourselves one of the bodies of a system. 

 We then perceive the reactions, and we have learned to correlate our 

 perceptions to the motions of the other bodies of the system. 

 Nevertheless , had we not possessed this extremely important sense, 

 we might have elaborated the same system of dynamics that we now 

 have merely by the sense of sight, as illustrated by the example of 



