OCTOBEK 5, 1917] 



SCIENCE 



341 



Mr. Patterson assumes a contradiction where 

 none exists and then proposes an artificial 

 way out. E. A. Eckhaedt 



Eandal Morgan Laboratory, 



PHILADELPHfA, Pa. 



THE THIRD LAW OF MOTION AND 

 " INERTIA REACTION " 



The recent article by Mr. Andrew H. Pat- 

 terson in Science for March 16, 1917, impels 

 me to add to the discussion of questions in 

 mechanics something that I have tried to make 

 clear to students. It is along the line of Mr. 

 Pulcher's article of November 24, and con- 

 cerns the confusion between the third law of 

 motion, the second law, and D'Alembert's 

 principle. 



Mr. Patterson appears to object to teaching 

 that " to every action there is always an equal 

 and contrary action " or that " forces always 

 occur in pairs " and at the same time that an 

 " unbalanced force " produces an acceleration. 

 There is surely no inconsistency in this, since 

 the " pairs " of forces or the action and re- 

 action act on different bodies, say A and B, 

 then if no other bodies are acting upon them, 

 there will be an unbalanced force on each, 

 and each will be accelerated, but in opposite 

 directions. Evidently another pair of forces 

 may act between B and C such that on the 

 whole the forces on B exactly balance, and yet 

 A will be left with an accelerated motion. 

 On the other hand, while it is clear from writ- 

 ing the equation representing the second law 

 of motion in the form F — Ma = 0, that 

 if a force equal to the mass times the acceler- 

 ation should act on the body in the opposite 

 direction to the impressed force, these forces 

 would be in equilibrium, this is not a case of 

 the third law, which specifies that the forces 

 considered act between two bodies and not 

 on one and the same body. If for a system 

 one adds the idea (D'Alembert's principle?), 

 that the internal actions and reactions of any 

 system of bodies are in equilibrium among 

 themselves, a special case of the third law, 

 one obtains the more general statement that 

 if forces equal to the several masses times 

 their respective accelerations were applied. 



etc., a form which is useful in the handling 

 of problems, but which does not imply that 

 such forces are acting and does not call for 

 the idea of " inertia reactions." 



The case where " inertia reaction " is most 

 frequently dra\^Ti in, in connection with action 

 and reaction is the instance of an object being 

 whirled around on the end of a string. Now 

 when one explains the motion of the moon 

 about the earth as due to the action of the 

 gravitational force on the moon directed tow- 

 ards the earth, one looks for the " reaction " 

 in a gravitational force on the earth directed 

 toward the moon, but not a force on the moon, 

 and this reaction on the earth has nothing to 

 do with the mass X acceleration of the moon, 

 but would be the same if the moon were at 

 rest in the position which it has at any in- 

 stant. Is not the same true for the ball and 

 string? Consider the case where a person 

 grasps the ball by a hook at the end of a di- 

 ameter, and pulls on a cord at the other end 

 with the force F, the ball as well as the cord 

 is strained, and we may say that the ball is 

 pulling on the string and the string on the 

 ball (the third law), in virtue of this strain. 

 Now let go at the one end, in order to continue 

 to apply a force F the hand must be moved 

 with the same acceleration which the ball has 

 in order to keep the string stretched, and 

 would not the ball in the neighborhood of the 

 string remain strained as before and hence 

 the forces between ball and string be of the 

 same nature as before ? Now suppose the ball 

 swung aroutnd the head, as !Mr. Patterson sug- 

 gests, would not the ball still remain strained 

 and would it not pull on the string with a 

 force which would be exactly the same as if 

 the ball were at rest, but in the same state of 

 strain? If so why bring in an inertia reac- 

 tion? In the illustration of the porter push- 

 ing a cart, as long as he actually pushes there 

 is an equal counter force on him, but in the 

 one case the push on the cart may be balanced 

 by friction, and in the other it would be an 

 unbalanced force on the cart. Actually if 

 friction suddenly ceased would not the porter 

 probably notice that the force with which he 

 was pushing had suddenly diminished, and 



