Beyond Newton and Archimedes

Contents: 

Chapter 1: 2360 Years Old Aristotle's Assertion Revalidated by Stokes Law • First Glimpse 1 • Introductory Mathematical Background of Aristotle's Assertion • Aristotle (384? 322BC) and his assertion • Violent motion or forced motion • Aristotle's assertion about falling bodies • In air • In water • Galileo's contradiction of Aristotle's assertion • Mathematical basis of Galileo's conclusions 8 • Distance travelled when the body moves with uniform velocity • Distance travelled when the body moves with uniform acceleration • Galileo's experimental demonstration • In vacuum • Merits and demerits of Aristotle's assertion and Galileo's demonstration • Mathematical equations based upon Archimedes principle, and falling bodies • Mathematical equations based on Archimedes principle • Can resultant weight be a measure of tendency to fall? • Re-validation of Aristotle's assertion by Stokes law • Stokes law • Constant velocity of small spheres • Calculation of terminal velocity or constant depending upon Stokes law • Revalidation of Aristotle's assertion • Motion of spheres of different radii (masses) • Confirmation of Stokes law • Frequently asked questions • References

Chapter 2: Construction of Water, Glycerine and Ethyl Alcohol Barometers • First glimpse • Atmospheric pressure • Glass tumbler experiment • The Magdeburg hemispheres • Some significant terms • First barometer constructed in 1644 • Measurement of pressure • Derivation of the equation of pressure or Pascal's law in presence of gravity • Role of surface tension, angle of contact and viscosity in measurement • Why the water barometer has not been constructed since 1644 • Formation of glycerine, water and ethyl alcohol barometers • Two methods to measure acceleration due to gravity • Newton's law of gravitation • Gravitation and weight • Variation caused by the acceleration due to gravity with altitude (height) • Comparison of two values of acceleration due to gravity • Calculations of heights of various liquid columns • Frequently asked questions

Chapter 3: Archimedes Principle: The Oldest Established Law • First glimpse • Archimedes • Origin of Archimedes principle • What was the unit of volume and weight at the time of Archimedes? • Relevant terms • Measurement of the density of body by the simplest and possibly oldest method • Measurement of relative density • Derivations of the Archimedes principle or Equation for the Archimedes principle (U = VDg) from its definition using the law of gravitation • Equation for the Archimedes principle (U = VDg) from pressure P = DgH • Numerical calculations of upthrust • Measurement of density by the method independent of the Archimedes principle • Laws of flotation • Exception to Archimedes principle or Factors not taken into account by the Archimedes principle • Frequently asked questions • References  

Chapter 4: The Generalized Form of Archimedes Principle • First glimpse • Introduction • Completely submerged floating balloons • The mathematical equation for the mass which the balloon supports while floating • Flotation of a completely submerged balloon or vessel or artifact in water • Similar equation in existing literature • Experimental confirmation of theoretical results • Experimental description • Experimental verification of Eq.(4.) • Possible reasons for deviations from Archimedes principle • Effect of density of additional mass attached • Possibility of significant deviations • The generalized form of the principle • 2264 years old Archimedes principle generalized • Generalized Archimedes principle and the completely submerged floating balloon or vessel or artifact • Recent developments • Frequently asked questions • References

Chapter 5: Prediction of Indeterminate Form Of Volume From Archimedes Principle • First Glimpse • Decrease in the weight of the body in a fluid (liquid or gas) • The equations in terms of acceleration due to gravity • General conditions of flotation • Equations for completely submerged floating balloons • Air-filled balloon floating in water • Calculations of variables, v, D_m, D_w and V based upon the original form of the Archimedes principle • Division by zero smoothly follows from equations based on the Archimedes principle • Numerical calculations of volume V • Generalization of the Archimedes principle • Mathematical derivation of the Archimedes principle is only true for symmetric shaped bodies. It is not true for body of any shape • Condition of derivation of upward thrust • Physical significance of the coefficient of proportionality • Floating bodies • Independent experimental support to above experiments • How the shape of the body is important in falling bodies • Accelerated motion • Downward motion of bodies when the Stokes law is not applicable • Frequently asked questions • References

Chapter 6: Is Stokes Law Applicable for Rising Bodies? • First glimpse • Viscosity • Newton's equation of viscosity • Stokes law • Derivation of F = 6p?rv • Determination of the magnitude of k • coefficient of proportionality • Constant velocity attained by a falling body • The three forces compete • Is Stokes law applicable to rising bodies? • Measurement of viscosity of fluids by the method of rising bodies • Mathematical background • A suitable body rising upward in a viscous medium • Measurement of viscosity of fluids by equation (6.31) • Frequently asked questions References

Chapter 7: Limitation of Existing Theories and an Alternate Theory of Rising, Falling and Floating Bodies • First glimpse • Relevant terms • Archimedes? principle • The Archimedes principle in rising, falling and floating bodies • Qualitative explanation • Quantitative explanation • Importance of resultant weight • The resultant acceleration follows from the resultant weight • In vacuum • Archimedes? Principle and quantitative displacements of bodies • Distance travelled by falling bodies in fluids • Motion of bodies in mercury ( 13600kg/m³) • Experiments in glycerine (12600 kg/m³) • Experiments in mercury ( 13600kg/m³) • Distance travelled by rising bodies in fluids • The Archimedes principle and floating bodies • First stage observations: Floating balloons in water • Factors not taken in account by the Archimedes principle • Stokes law • Constant velocity attained by the falling body • Subtle range of Stokes law • Drag force • Developments after the Stokes law • Rigorous requirement of generalized theory • Background of a generalized or complete theory • Force exerted by fluid or medium Fm • Understanding effect of xm • Understanding effect of ym .0. Force exerted by the body (F_b) on the medium • Understanding the effect of x_b • Understanding the effect of y_{b •} Units and dimensions of a_b and a_m .? Comparative study of F_m and F_b or hidden ratio • Explanation for motion of bodies • Standard conditions on the basis of alternate theory • The displacement in terms of the falling factor and the rising factor • Comparison of equations giving distances (upward and downward) • The floating bodies • The falling bodies and generalized theory • Standard conditions • The rising bodies and alternate theory • The floating bodies and the generalized theory • Conclusions • References

Chapter 8: Route to Newton's Laws of Motion • First glimpse • Aristotle and his dynamics • The theory of cause and effect • Motion of a projectile • Johannas Philoponus's theory of impetus • Papal decree and theory of impetus • Jean Buridan coined impetus • Galileo's law of inertia • Demonstration of Galileo's law of inertia • Linear momentum • Force • Newton's three laws of motion • Three laws of motion in Latin and English • First law of motion • Explanation by Newton of the first law of motion in the Principia • Second law of motion • Mathematical expression of Newton's second law of motion • Second law of motion as taught in textbooks is not given by Newton • Definition II in the Principia Book 1 • Newton discovered the calculus and did not use derivative of time, i.e. d/dt • Who defined Newton's second of motion in the form F = kma? • Are p = mv and F = ma analogous • ? F = ma does not follow from Newton's second law of motion • The value of the coefficient of proportionality K in F = Kma is determined assuming values of all variables are equal to unity • Coefficient of proportionality G in law of gravitation • Arbitrary method of determination of G is not justified • The value of the coefficient of proportionality • can be measured by this method • Why special status for F = Kma? • Newton's Principia and the third law of motion • Action and reaction are not always equal and opposite • Following example is neglected • Second law of motion is the real law of motion • Prediction of undefined inertial mass from equation

F= ma or m= F/a • Inertial mass under the conditions when Newton's second law of motion reduces to first law of motion (a=0, u= v, F= 0) • Core area of applicability of Newton's first and second laws • The facts Newton did not explain • Two types of systems • Frequently asked questions • References

Chapter 9: Experimental confirmations of equations of conservation laws in elastic collisions • First glimpse • Laws of conservation in nature • Linear momentum • Principle of conservation of linear momentum • Questions to be pondered upon • Work • Units and dimensions of work • Kinetic Energy • Newton's first law of motion or the law of conservation of velocity • Newton's first law of motion or the law of conservation of velocity • Newton's first law of motion, law of conservation of momentum and law of conservation of kinetic energy • Collisions • Elastic collision • Inelastic collisions • Perfectly inelastic collisions • Coefficient of restitution or coefficient of resilience • Final velocities of bodies v₁ and v₂? The values of v₁ and v₂ under special conditions • When two bodies are of equal masses • Masses of bodies are different and body B (target) is at rest, u₂=0 • When mass of body B is negligible compared to that of body A (M₂ << M₁) • Experimental confirmation of the equation based upon conservatism laws • The general values of u and v • When two bodies are of equal masses, M₁= M₂= M • When target is at rest (u₂=0) • Large difference in velocities of body A and body B • Concluding remarks • Experimental aspects • Frequently asked questions

Chapter 10: One-dimensional elastic collisions and Newton's third law of motion • First glimpse • The third law of motion as in the Principia • Newton's original explanation to third law of motion • Examples which illustrate the law qualitatively • The following example is neglected • Elastic collision in one dimension: mathematical equations • When body B is at rest (u₂=0) • Mass of body B is very-2 large compared to mass of body A • Generalised form of the third law of motion • Frequently asked questions • References • Index