Newton s Third Law of motion Generalized

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1 1 Newton s Third Law of motion Generalized Ajay Sharma Fundamental Physics Society His Mercy Enclave Post Box 107 GPO Shimla HP India ajay.pqrs@gmail.com website Abstract Newton s third law of motion have been quoted from the Philosophiæ Naturalis Principia Mathematica published in The law as in original form states that to every action there is always an equal and opposite reaction: or the forces of two bodies on each other are always equal and are directed in opposite directions. The equality of action and reaction is unconditional or unrestricted or uninfluenced by external factors in the law i.e. equality holds good in all cases. This is discussed here at macroscopic level and some anomalous results are found due to characteristics of the processes.

2 2 1.0 Newton s Principia and laws of motion Newton s most famous book popularly known as the Principia was written originally in Latin as Philosophiæ Naturalis Principia Mathematica in It was translated to English by Andrew Motte in 1729 as "Mathematical Principles of Natural Philosophy". Two full English translations of Newton's 'Principia' have appeared, both based on Newton's 3rd edition of Newton three laws of motion in original Latin and English are given below. Newton s third law of motion as stated in original Latin Lex III: Actioni contrariam semper et æqualem esse reactionem: sive corporum duorum actiones in se mutuo semper esse æquales et in partes contrarias dirigi. When translated to English Law III: To every action there is always an equal and opposite reaction: or the forces of two bodies on each other are always equal and are directed in opposite directions. Newton s laws of motion as stated in original Latin Newton explained the law in the following way in the Principia. Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone. 2.0 Action and reaction are not always equal and opposite Consider a body A of weight W is lying over the surface of the other body B. The body A exerts force W on the other body B. Thus, Action of body A = W According to Newton s third law of motion the body B, exerts equal and opposite reaction R to body A Reaction of body B =R Now both acting and reacting body remain at respective positions, thus Action of body A = -Reaction of body B or R = -W (1)

3 3 If in general body A exerts force F AB on body B, then body B exerts force F BA on body A then according to Newton s third law of motion F AB =-F BA (2) Thus action and reaction are equal and opposite. Also there are some qualitative examples for understanding the third law of motion. For example a swimmer pushes water backward and moves in the forward direction. It easier to walk on the earth as we get sufficient reaction. On the other hand it is difficult to walk on the sand as adequate reaction is not obtained. A ball is hit on the wall then as a reaction it rebounds. In elastic collisions if the target is of sufficiently large mass, then as a reaction the projectile rebounds. A can be general law if it is quantitatively confirmed in all cases where it is applicable. 2.1 Following example is neglected Consider a boy is standing in front of concrete wall, holding a rubber ball and cloth ball in the hands. Let the wall is at the distance of 10m from the boy. First case: Let the boy throws the rubber ball at the wall with force F. The ball strikes the wall, and comes back to the boy i.e. travelling 10m. Now action and reaction can be understood as Action: Boy pushes the ball towards wall Reaction: The wall pushes the ball towards boy. As the boy pushed the ball from distance of 10m, and wall rebounds to distance of 5m. So action and reaction are equal and opposite. Boy pushes the ball through 5m = Wall rebounds the ball through 10m Action = -Reaction (3) The ve sign indicates that after reaction ball moves in opposite direction. In this case Newton s third law of motion completely obeyed. Second case: Let the boy throws the cloth ball at the wall which is at distance of 10m. Let the boy throws the cloth wall with same force F. The ball strikes the wall, and rebounds to only 3m. Now action and reaction can be understood as Action: Boy pushes the ball towards wall Reaction: The wall pushes the ball towards boy As the boy pushed the cloth ball from distance of 5m, and wall returns to distance of 3m. Thus boy pushes the ball towards the wall through 5m

4 4 The wall rebounds the ball through 3m Action Reaction So action and reaction are opposite but not equal in case of cloth ball. In addition, the quantitative confirmation of mathematical equations based upon elastic collisions (when mass of target is sufficiently larger than projectile) will also lead to concrete results in this regard. The experimental verification of such equations include measurement of rebound velocities of projectiles. There can be many such examples. It is just possible that characteristics of ball and wall are such that ball may rebound more than 5m. Conclusions: Thus reaction also depends upon various characteristics of body or ball (cloth, rubber) and wall (concrete or wooden or earthen). Thus reaction depends upon various characteristics of the system. In case of elastic collision ( as cited above), the natures of projectile, target and surface will play a significant role. There can be many such examples. The quantitative confirmation of mathematical equations based upon elastic collisions (when mass of target is sufficiently larger than projectile) will lead to concrete results in this regard. According to definition of the third law of motion, the law is absolute or unrestricted i.e. or the forces of two bodies on each other are equal and are directed in opposite directions. However here forces acting the ball and wall are opposite but not equal. The inherent characteristics of the process play significant role. Thus the third law of motion is generalized. Generalization: It allows us to reassess Newton s third law of motion as To every action there is always a opposite reaction which may or may be equal depending upon characteristics of the system. Reaction Action or Reaction = -k Action Where k is coefficient of proportionality which depends upon characteristic conditions of the process. If k=1, then Reaction = - Action Thus the third law of motioned generalized in case of macroscopic bodies. Some suggested experiments will further be useful in understanding the law. Acknowledgement

5 5 Author is highly indebted to Dr Stephen Crothers, Dr R Stanley and Anjana Sharma for encouragements and critical discussions. References 1. Newton, Isaac Mathematical Principles of Natural Philosophy pp.19-20, London, 1727, translated by Andrew Motte from the Latin.

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