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“BEYOND NEWTON AND ARCHIMEDES”

Home Page http://www.cisp-publishing.com/acatalog/info_109.html

Publisher Cambridge International Science Publishing, Cambridge, ENGLAND

Pages 330, Date of Publication Oct. 2013. No of Chapters 10

About Author

Ajay Sharma is working as Assistant Director in Department of Education, Shimla (India). He is author of book Beyond Newton and Archimedes. He began his career as a physics lecturer at DAV College Chandigarh (India).In the book he has generalized Newton’s laws and 2246 years old Archimedes principle. He maintains Newton did never give F =ma but it was speculated by Eular in 1750. Archimedes has been generalized as it does not account for the shape of body. He stresses formation of water barometer as it provides alternate method for measurement of g. It has been done for first time in the history of Science. His second book Beyond Einstein and E=mc2 is in press. He has published over 45 research papers and invited to over 85 international conferences. Mobile 0091

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9418450899, Email ajay.pqrs@gmail.com His work is available online at: http://www.AjayOnLine.us

forth coming book Beyond Einstein and E=mc2

Description of 10 Chapters

Chapter 1: 2360 Years Old Aristotle's Assertion Revalidated by Stokes Law First Glimpse

According to Aristotle’s assertion of falling bodies, ‘heavier body falls down quickly’. It got immediate support from the fact that a stone falls quickly than a straw. Mathematically, falling tendency mass of body.

In 1851,Stokes put forth that under certain conditions the bodies attain constant velocity. According to relevant mathematical equations, Average velocity mass of body

It was contradicted by Galileo [1564-1642] after 2000 years of use stating that all bodies fall with same acceleration i.e. travel equal distances in equal intervals of time. It got immediate support from the fact that a ten pond shot and one pond shot fall at same time in air. Thus

‘all bodies fall equal distances from equal intervals of time.’ Mathematically

S= at2

where a is acceleration

Comparing Aristotle’s assertion and Stokes law , it is obvious that both have similar forms. Thus Aristotle’s assertion is as useful as Stokes law in case of falling bodies.

If the resultant weight is regarded as falling tendency , thenFalling Tendency massThus in this case also Aristotle’s assertion is justified. Acceleration is required to calculate the distance travelled by body.

(i) According to Aristotle’s assertion of falling bodies, ‘heavier body falls down quickly’.(ii) It got immediate support from the fact that a stone falls quickly than a straw.Galileo [1564-1642] after 2000 years of use contradicted Aristotle’s thesis and stated that ‘heavier and lighter bodies fall at same rate’For example a one pond and ten pond shot fall at same rate. It is presciely true in vacuum only.(iii) English Physicist George Grablei Stokes, studied effect of viscosity in small spheres and found that they move with constant velocity.The mathematical equation relates constant velocity with mass

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v =

v =ZM thus ‘Heavier spheres (range in which Stokes Law holds good) fall quickly than lighter ones’.Thus Aristotle’s assertion is valid in fluids ( narrow range) even when Stokes law is valid.Hence abundoned Aristotle’s assertion, is valid upto some extent.

Chapter 2 Construction of Water, Glycerin and Ethyl Alcohol Barometers First Glimpse

Torricelli had made mercury barometer in 1644 and measured pressure as P=DgH, but in past 370 years the water barometer has not been formed.

The height of water column in water barometer must be 10.33 m.

Does viscosity of liquid (not taken in account till date) has any effect on height of liquid in the barometer?

Will results of glycerine and water barometers differ from mercury barometer?

The value of acceleration due to gravity, g can be measured by this method and used in measurement of mass of earth and other astronomical data.

If both methods give different values of g, then what would be mass of earth? It is a big question.

If experiments fail then it would lead to reanalysis of equation P =DgH

The value of acceleration due to gravity, g varies with altitude and depth , it can be compared with g =P/DH. Thus value of g can be cross checked.

The value of acceleration due to gravity varies with altitude and depth, it can be compared with P =Dgh. All values must be consistent.

(i) Torricelli had made mercury barometer in 1644 and measured pressure as P=DgH, the equation

P=DgH, became derivable after 1685. The height of mercury column in the barometer is 76cm or 0.76m

of mercury is estimated with P=DgH.

(ii) Water is the most abundant liquid but scientists have not constructed the Water Barometer in past

370 years. The height of water column mathematically comes out to be 10.31m

Height of water column = 10.31m (P/Dwg)

Similarly

Height of ethyl alcohol column = 13.6 m (P/Deag)

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Height of glycerin column = 8.204 m (P/Dglg)

The comparison of experiments related with water and glycerin is interesting, as coefficient of viscosity

of glycerin is 1058 times more than that of water. Whereas density of glycerin is only 1.26 times density

of water. The role of viscosity is expected to be significant.

(iii) Importance: Due to intricacies of the experimental and theoretical status the height of water

column may not be 10.31m ( say it is 10.8m), then real results will be on the hand. It would mean the

value of g (acceleration due to gravity) will vary (P = DHg). Thus mass of the earth

M = gR2/G

will vary (current value of mass of the earth (M=5.974x1024 kg) due to variation in g . Likewise distances

between various heavenly body and their masses will vary. Thus simple looking undergraduate level

water barometer experiments are very significant. These barometers are not formed in history of

science For complete understanding all barometers must be formed so that results may be drawn over a

wide range. It is established in science that conclusions are not drawn from a single observations. Many

editors and scientists of journals have asked me for experimental results, as method is okay.

3. Archimedes Principle: The Oldest Established Law

First Glimpse

General public all over the world knew before Archimedes about the observation that a body pulled from water, then it is lighter.

According to anecdote or legend that Archimedes enunciated the principle, to ascertain the purity of king’s crown but there is no documentary proof for this belief. Also at that time there was simpler method to determine the density of crown.

At time of Archimedes, the unit of volume such as one cubic centimeter and one milliliter was not defined. Also mass was not defined as we do now. Thus measurement of density may not be as accurate as now.

It is strange that mathematical equations were formed on the basis of Archimedes principle (250BC) after 1935 years of its enunciation i.e. in 1685 when Newton published The Principia and defined acceleration due to gravity ‘g’.

Archimedes principle implies that upthrust ( U=VDg) depends upon volume of body V, density of medium D and acceleration due to gravity, g.

The upthrust experienced by body does not depend upon the SHAPE of body.Thus upthrust ( U=VDg) for bodies of same volume is same irrespective of shape .

The effect of shape of body on upthrust can be studied experimentally in various cases i.e. rising, falling and floating bodies.

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The magnitude of units of mass and volume have changed time and again in the history of science, so it is one’s guess how density is measured precisely? Thus density of substances varied in past. So which value Archimedes confirmed.

(i) General public all over the world knew before Archimedes that a body is pulled from water, then it is

lighter i.e. weight decreases. Archimedes stated that …… how much weight decreases in fluids?

Archimedes stated the principle in 250BC i.e. 2265 years ago.

(ii) There is no documentary proof that Archimedes stated the principle to determine the purity of king’s

crown. It is an anecdote only.

(iii) 2265 Archimedes principle simply involves determination of volume. But the unit of volume (1cc)

was defined in 1879 i.e. after 2129 years after enunciation of the principle. The value of 1cc was

defined in 1889, then redefined in 1907 and again changed 1964. So volume was not uniformly defined.

Thus Archimedes principle is either wrong now or was in the past.

(iv) None of the original writings of Archimedes exist now. There are books which describe works of

Archimedes.

4. The Generalized form of Archimedes principle

First Glimpse

Archimedes stated the principle in 250BC i.e. 2265 years ago. It is the oldest established principle in science.

Archimedes principle has not been confirmed in completely submerged floating balloon experiments in water and other fluids? In these experiments only weight and upthrust act on the balloon.

Mathematical equations based upon the principle became feasible after 1935 years ago i.e. in 1685 when Newton published the Principia. In the book Newton defined Law of Gravitation and acceleration due to gravity, g.

The rising (hot air) balloons were discovered in 1783 by Montgolfier brothers.

From mathematical equations it is evident that the mass [as it appears in equations i.e. mf =(V+vf) Dw-VDa ] which balloon supports is independent of shape of balloon.

Till date in the history of science no specific experiments are conducted to check effect of shape of completely and independently submerged floating balloon or bodies.

Such experiments can be conducted in viscous liquid glycerin and the dense liquid mercury (13,600kg/m3).

The generalized form of Archimedes principle is U=fVDmg, where f is coefficient of proportionality.

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f takes in account the factors like shape of body, viscosity of medium etc. , which are not taken in account by Archimedes principle.

(i) Mathematical equations based upon the principle became feasible after 1935 years ago i.e. in 1687

when Newton published the Principia. In the book Newton defined Law of Gravitation and acceleration

due to gravity, g. How the principle was regarded as true without mathematical equations for 1937

years, it is anyone’s guess? The weight, buoyant force and resultant weight were defined. Thus

quantitative explanation became possible. The principle is applied in explaining the rising, falling and

floating bodies. Such equations became feasible 327 years before even then resultant acceleration of

bodies in fluids is not calculated, hence distance travelled in specific time is not calculated. It is done for

first time. Similar is case of rising bodies.

(iii) The critical analysis also leads to interesting results in floating bodies. By floating bodies (completely

submerged) we mean , bodies float under actions of weight and upthrust alone. In case of completely

submerged floating balloons or bodies

weight of body = upthrust or VDbg = VDmg or Db =Dm

Thus Archimedes principle takes only densities (Db , Dm) in account but not following factors.

(a) Shape of body : Floating balloon may be umbrella shaped, needle like , cone : the floating effects are

same in all cases.

(b) Viscosity of fluid: The glycenine has viscosity 1058 times that of water and density of glycerin is

1.26 times that of water that of medium, so experiments can be conducted in it. This effect is taken in

account.

Experiments have been suggested to determine the effects of shape of body and viscosity of medium.

(iv) If the 2265 years old Archimedes princilple is generalized then additional coefficient of

proportionality , appears in mathematical equations. This coefficient takes in account the shape of

body, viscosity of medium , other factors etc.

(v) Mathematical equations

Original form of Archimedes principle

U =VDg

U : upthrust , D density , g is acceleration due to gravity

The generalized form of Archimedes principle ( upthrust is prportional ro weight),

U =fVDg

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where f is coeficient of proportionality , it can be detrmined experimentally like numerous others

existing in physics.

5. The Prediction Of Indeterminate Form Of Volume From Archimedes Principle

First Glimpse

2265 years old Archimedes principle is undoubtedly regarded as the oldest established law.

At time of Archimedes mathematical equations involving g did not exist, so it was not completely analyzed mathematically by Archimedes.

Even concepts of viscosity and surface tension did not exist at time of Archimedes.

Consequently when Archimedes principle (U =DmVg) is mathematically analysed then some inconsistent results are obtained. For example the volume of body becomes UNDEFINED i. e V

= which is meaningless.

Thus the oldest established principle is generalized for first time i.e.

‘when a body is wholly or partially immersed in fluid at rest then it experiences upthrust and its weight decreases which is proportional to the weight of fluid displaced.’Mathematically, U = f DmVg

When generalized Archimedes principle, U = f DmVg is used then exact volume is obtained i.e. V =V.

Some experiments are required to confirm the generalized form of Archimedes principle. f is coefficient of proportionality, it depends upon factors like shape of body , coefficient of

viscosity etc. which are not taken in account by Archimedes principle.

The mathematical equations became possible about 330 years ago, when Newton defined acceleration

due to gravity g. These equations (many more in the following discussion ) are not critically analyzed till

date. When I critically analyzed these , then inconsistent results are formed.

(i) There is peculiar prediction from Archimedes principle which is pointed out for first time.

Consider a floating (completely submerged) balloon in water or other fluid. Under some conditions the

volume of medium filled in balloon (200cc, platinum, silver, cork, etc.) mathematically becomes

V = 0/0

The eq.(5.18) gives value of volume, V.

V = (5.18)

Now substituting eq.(5.19) and eq. (5.20) i.e. Dm =Dw in eq.(5.18 )

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V= = = (5.27)

or Volume of platinum, silver, cork, metal and air =

It is undefined, the fractions 3/8 , 4/9 etc are only defined , not 0/0. Hence limitation.

(ii) If we use the generalized form

U =fVDg

Then we get

V=V

which is true. Hence generalised form of Archimedes principle is justified.

Now substituting eq. (5.34) in eq.(5.35) and Dm=Dw we get

V= = =V (5.37)

The values of f can be calculated for body of arbitrary shape floating in the glycerine ( coef. Of viscosity for glycerine is 1058 times than that of water, and density is only 1.26 times that of water).

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6. Is Stokes Law Applicable for Rising Bodies? Pp.137-154

All bodies fall in vacuum with maximum acceleration g (9.8m/s2 )in vacuum . According to Archimedes principle bodies fall with reduced acceleration in fluids.

Stokes law (1845) predicts that body falls in fluids with subtle constant velocity, under some conditions. This mathematical background is used in measurement of viscosity of fluids by method of falling bodies.

The viscous force is given by F= kηrv= 6πη rv =18.8571ηrv.

Sir George Gabriel Stokes (1819 –1903), Lucasian Professor of Mathematics did not experimentally measure the value of k equal to 6π . He might have given it theoretically.

Stokes law is not confirmed for all materials and liquids and apparently value of k as 6=18.8571 is determined theoretically. Thus its study is incomplete.

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Stokes law is regarded as true for rising bodies when v is negative but conditions (like postulates in falling bodies) of applicability of the law have not been ascertained theoretically or experimentally.

If experimental conditions of applicability are ascertained, then viscosity of fluids can be measured by method of rising bodies.

Without experiments regarding falling and rising bodies over wide range of parameters the basic law of fluid dynamics is incomplete.

Stokes law holds good for falling bodies under five assumptions. It is experimentally confirmed by Arnold

for spheres of rose metals of radii 0.002cm in water. It is used to determine the viscosity of fluids. The

small sphere attain constant (terminal ) velocity due to viscous force,

F=kηrv=6πη rv=18.8571η rv (6.14)

The equation for terminal velocity is given by

v= (6.21)

v is constant velocity, r is radius of small sphere, Db density of body , Dm density of medium , g is

acceleration due to gravity and η is coefficient of viscosity.

Next question is whether Stokes law holds good for rising bodies as well . If yes, then what are

conditions it holds good ? If under some conditions the rising body attains constant velocity then

v = (6.36)

There is no information in existing literature on the topic. It has to be investigated. The study will throw

light on the motion of rising bodies .This chapter critically analyze the state of rising bodies, which is not

studied at all in the existing literature. Experiments must be conducted to determine the region of

validity of Stokes law is discussed. It is basic problem in fluid dynamics and is incomplete.

7. Limitations of Existing Theories and an Alternate Theory of Rising, Falling and Floating Bodies

pp.155-220

First Glimpse

Here natural motion of bodies (fall or rise without external influence) is considered. The bodies fall under influence of gravity.

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In existing physics Archimedes principle and Stokes law are used to explain the motion of rising, falling and floating bodies.

Archimedes principle states that if Db>Dm body rises, Db<Dm body falls and if Dm=Db then body floats. It is qualitative explanation only.

The principle does not correctly points out the ‘how much distance’ is travelled (fallen or risen) by body in time t i.e. 1s, 4s or 9s etc.

According to Archimedes principle bodies of aluminum (2700kg/m3 ) of mass 1mg ( spherical in shape) and 1000kg ( flat in shape ) must fall through distance of 3.085 m in 1s. But this prediction is not confirmed yet, but regarded as true.

The principle only takes in account the Dm and Db, and neglects mass, shape and angle at which body is dropped, magnitude, characteristics motion of medium and convectional currents etc.

R Piazza has observed anomalous observations to Archimedes principle in sensitive experiments, that heavy particles of gold floated over the surface of lighter medium.

Stokes law is applicable in very-2 narrow range i.e. Arnold confirmed it for bodies of rose metal of radii 0.002cm with some accuracy, otherwise it is not valid.

The drag force is not applicable to natural motion. Thus an alternate theory on rising, falling and floating is developed taking all factors in account

(i.e. mass, shape and angle at which body is dropped, magnitude, characteristics motion of medium and convectional currents etc.)

The newly discussed terms are Hidden Ratio, Falling Factor and Rising Factor.

(i) If we drop a 10kg steel ball, 1kg steel cylinder, 1mg steel sheet in water. In the existing

literature, there is no equation which may predict that in 10s, how much distances bodies will

travel?

According to equations based upon Archimedes principle, all bodies must fall equal DISTANCES

in equal interval of TIME. It is based upon following equations

The resultant weight is the difference weight and upthrust [6].

WR =W-U = (Db-Dm)Vg

= (1- )VDbg (7.11)

or g =

Thus dividing eq.(7.11) with mass of body, m =DbV,

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aR = (1- )g

If body starts from the rest i.e. u=0

S = (1- )gt2 (7.24)

Thus distance travelled is independent of mass, shape of body etc. But is not justified

experimentally, even in daily life observations.

(ii) The same is true for rising bodies. These phenomena ( distances travelled in certain times)

are not discussed in existing literature. The mathematical equations don’t exist in literature, and

hence has been purposely formed. These equations must be experimentally confirmed.

Regarding motion of bodies other prevalent concepts such as viscous force

F= 6 ηrv (7.51)

D=CDm Au2 (7.58)

and Tchen’s equations are discussed. All the theories in the existing physics don’t explain the

phenomena of rising and falling bodies QUANTITATIVELY. These have different applications.

(iii) An alternate theory is developed, taking all possible factors in account e.g.

magnitude of medium, shape of medium, state of motion of medium, convectional currents,

viscosity of medium , surface tension of medium , fluidity of medium , magnitude of body,

shape of body, distortion of body and angle at which body is dropped. New equations are

formulated. All the phenomena are discussed with help of mathematical equations.

Experiments are required to confirm them.

8. Route to Newton's Laws of Motion pp.221-263

First glimpse

Aristotle put forth that ‘there can be no motion without force’ or ‘cause and effect perception’, but was unable to explain the projectile motion.

Philoponus in 6th century and Buridan in 14th century contradicted this assertion and put forth Impetus Theory.

Galileo hypothesized a medium devoid of resistive forces. In early 17th century Galileo preached the law of Inertia that body maintains its uniform motion

even without force if once set in motion. Apparently, it was just opposite to Aristotle’s assertion and Buridan/Philoponus Impetus Theory.

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Whereas Aristotle, Philoponus and Buridan had maintained that force or impetus ( energy imparted) is required for motion; they considered medium with resistive forces.

In 1687 Newton re-stated Galileo’s law of inertia and gave the laws of motion, in treatise the Principia.

Galileo’s law of inertia and Newton first law of motion imply that as long as resistive forces (atmospheric, frictional, and gravitational etc.) are absent body moves with uniform velocity and stops when resistive forces start.

It would be better if scientists formulate, a comprehensive law for practical system ( having resistive forces) and this law must reduce to ideal cases ( devoid of resistive forces), such as Galileo’s law of inertia.

(i) Origin of Newton’s laws lies in doctrines of Aristotle ( 383-322), Philoponus (520-600 ) , Buridan

((1295-1360) and Galileo (1564-1642 ). Newton (1642-1527) based his first law on the existing basis.

The various aspects are discussed with details.

The most important significance of the chapter is that

‘In the form of Newton’s second law is taught is schools (F=ma) all over the world , was not given by

Newton. It was given by Leonhard Eular in 1950.’

Definition of second law

“The alteration of motion is ever proportional to the motive force impress'd; and is made in the

direction of the right line in which that force is impress'd.”

Newton has defined motion or Absolute motion at page 10 of the Principia, as velocity.

Body is in motion if v>0 , alteration = simple difference

It means F =(v-u) not F=m (v-u)/t or F =ma , F force , m mass and a is acceleration

(i) In 1716 Jacob Hermann in his book “Phoronomia”, ( even now it is in LATIN) , gave equation

dc = p dt, where stands for “celeritas” meaning speed, and stands for “potentia”, meaning force.

Force = rate of change of speed, ( it is acceleration )

It is regarded as F=ma but it is definitely not , as it is acceleration but called force (potentia in Latin).

Newton’s Principia used “celeritas” as speed.

(ii) Momentum was defined in J Jenning's Miscellanea in 1721 as QXV (mass x velocity)

Newton ignored it when published last edition of the Principia in 1726

(iii) It must be noted that term impetus (similar to momentum) was defined by Jean Buridan (1295-

1360)

Impetus = Weight x velocity (mass x velocity) ….. momentum

(vi) Further in two significant papers entitled ‘Recherches sur le mouvement des corps célestes en

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général’ and then in Découverte d’un nouveau principe de Mécanique .

Euler used an extrinsic references frames (a system of three orthogonal Cartesian axes) and formulated

the second law of motion in this way:

2Mddx=Pdt2, 2Mddy=Qdt2, 2Mddz=Rdt2,

or P = , Q = , R =

where M is the mass and P, Q, and R the components of the force on the

axis (the coefficient 2 depended on the unit of measure).

Thus above equations represent F=ma = m , which is taught now as mathematical form of Newton’s

Second law of Motion.

EULER, Leonhard Recherches sur le mouvement des corps célestes en général, Mémoires de l’académie des sciences de Berlin 3, 93-143 or Opera, ser. 2, vol. 25, 1 – 44 (1747).

EULER, Leonhard (1750) “Découverte d’un nouveau principle de Mécanique”, Mém., 6, 185-217 or Opera, ser. 2, vol. 5, 81 – 108, (1750).So it is confirmed that Newton did not derive F=ma it was derived by Leonhard Eular in 1750.

(vii) Newton’s second law ( as given by Eular) predicts UNDEFINED MASS

Second law of motion (F=ma ) reduces to First Law of Motion ( body keeps state of rest or uniform velocity, a =0 when no force acts on body, F=0) under the condition when no force acts on system i.e. F =0 and a=0 Under this condition inertial mass ism =F/a = 0/0which is undefined. Eular should have discussed this aspect. There can be no limitation bigger than this. It is unnoticed by scientific community.

9. Experimental Confirmations of Equations of Conservation Laws in Elastic Collisions.

The equations based on elastic collisions take in account directly mass, but not other characteristics of bodies and experimental set up.

The important thing is to experimentally confirm equations.

If a projectile is very-2 heavy than target, then after collision target must move with double velocity.

When two bodies of equal masses collide, then they exchange their velocities.

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If the equations of conservation of momentum and kinetic energy are written but not experimentally confirmed, then job is half done.

It is stressed that at macroscopic level the conditions under which these equations hold good must be determined.

It can be experimentally checked up to which extent composition of bodies and resistive forces of the system play significant role in collisions.

The study of elastic collisions (conservation of momentum and kinetic energy) is incomplete if equations are not experimentally verified.

Mathematical equations based on elastic collisions predict that velocity of projectile and target after collisions is uniform i.e. acceleration is zero.

The conditions under which the conservation laws hold good, need to be established.

(i) Let a shell of mass 0.01kg is fired from the gun of mass 25kg. If muzzle velocity of the shell is 200 ms -

1, what is recoil speed of gun?

Mathematically speed of gun turns out to be -8cm/s . It can be determined under which conditions the

gun recoils with velocity 8cm/s.

We have equation for one dimensional elastic collisions

Or v1 = (9.25)

v2 = (9.29)

The equations have various sub-cases. The experimental set up is described to confirm above equations

experimentally.

10. Elastic Collisions in One Dimension and Newton's Third Law of Motion

In the Principia

First Glimpse

Newton quoted three laws of motion in the Principia very briefly. These laws are quoted at page nos. 19-20 of the first English translation of The Principia by

Andrew Mott in 1729.

The third law of motion is understood qualitatively at macroscopic level in standard textbooks.

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The third law of motion can be justified on the basis of equations based upon elastic collisions, especially when target is very-2 heavy than projectile.

The real beauty of science is to experimentally confirm the mathematical equations.

In view of discussion it is concluded that “To every action there is reaction but may or may be equal depending upon characteristics of the system.”

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 opposed an equal reaction: or the actions of two bodies upon

each other are always equal, and directed to contrary parts.

Limitations of 3rd law of motion (striking of ball on the wall).

Third Law: It is unconditional, completely independent of characteristics of interacting bodies i.e. it acts

equally for rubber ball, cloth ball, flexible ball, mud ball etc.

If these balls are hit on concrete ball with same force (same action), then must rebound to same extent.

But it does not happen, so action and reaction are not equal.

Table I Comparison of action and reaction on rubber and cloth balls (soft, softer, flexible, rigid or

typical) when hit on the wall.

Sr. No Bodies Action Reaction 3rd Law of Motion

1 Rubber ball F=2N

(10m)

F=2N

(10m)

Action =-Reaction

2 Cloth ball

(soft, softer, flexible,

rigid or typical)

F=2N

(10m)

F=1N

(5m)

Action -Reaction

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Author Ajay Sharma

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Email ajay.pqrs@gmail.com Mob. 0091 94184 50899