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Marlyne Montero Villareal MSciEd Physics II July 16, 2011 Mass

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Make Physics as simple as possible, but no simpler. Albert Einstein The ideal is to reach proofs by comprehension rather than by computation. Bernhard Riemann

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N EWTON S T HEORY OF M ECHANICS AND G RAVITATION Inertial Mass – the ratio between force and acceleration in Newtons Second Law and thus measure a particles resistance to acceleration Gravitational Mass-This is defined by the force of gravitation, which states that there is a gravitational force between any pair of objects, which is given by Where G is the gravitational constant. Wolfgang Rindler Relativity: Special, General and Cosmology 2 nd Ed. (2006)

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M ASS IN THE S PECIAL R ELATIVITY

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3 Conservation Principles: 1.Conservation of Energy 2.Conservation of Mass 3.Conservation of Momentum

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Relativistic Energy 2 Conditions: 1.The total kinetic energy E of an isolated system is conserved 2.E will approach the classical value when u/c approaches zero. Paul A.Tipler & Ralph A. Llewellyn Modern Physics 2 nd Ed. (1999

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Paul A.Tipler & Ralph A. Llewellyn Modern Physics 2 nd Ed. (1999

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Relativistic total energy is then defined as the sum of KE and the rest energy Paul A.Tipler & Ralph A. Llewellyn Modern Physics 2 nd Ed. (1999 )

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rest mass (velocity-independent) relativistic mass (velocity-dependent)

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Guide questions: 1.Does mass depend on velocity? 2.Is mass invariant? 3.Should using the term relativistic mass be discontinued?

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Does mass depend on velocity, dad?

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The notion of the dependence of mass on velocity as introduced by Lorentz in 1899 and then developed by him and others in the years preceding Einsteins years. Transverse mass; F is perpendicular to v Longitudinal mass; F is parallel to v Lev B. Okun The Concept of Mass Physics Today. June 1989

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In accordance with the theory of relativity the kinetic energy of a material point of mass m is no longer given by the well-known expression but by the expression This expression approaches infinity as the velocity v approaches the velocity of light c. The velocity must therefore always remain less than c, however great may be the energies used to produce the acceleration Albert Einstein The Special and General Theory of Relativity.1920

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If we develop the expression for the kinetic energy in the form of a series, we obtain The first term does not contain the velocity, and requires no consideration if we are only dealing with the question as to how the energy of a point-mass depends on the velocity. The most important result of a general character to which the special theory of relativity has led is concerned with the conception of mass. Before the advent of relativity, physics recognised two conservation laws of fundamental importance, namely, the law of the conservation of energy and the law of the conservation of mass; these two fundamental laws appeared to be quite independent of each other. By means of the theory of relativity they have been united into one law Albert Einstein The Special and General Theory of Relativity.1920

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Albert Einstein The Special and General Theory of Relativity.1920

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Albert Einstein The Special and General Theory of Relativity.1920

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Albert Einstein The Special and General Theory of Relativity.1920

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Albert Einstein The Special and General Theory of Relativity.1920

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Arthur Beiser Concepts of Modern Physics, 6 th Ed. (2003)

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The total energy and momentum are conserved in an isolated system, and the rest energy of particle is invariant. Arthur Beiser Concepts of Modern Physics, 6 th Ed. (2003) Paul A.Tipler & Ralph A. Llewellyn Modern Physics 2 nd Ed. (1999

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Four-vector In relativistic energy and momentum we components of another four vector The magnitude of the invariant energy/momentum four-vector is the rest energy of the mass m. Thus, observers in all inertial frame will measure the same value for the rest mass. Paul A.Tipler & Ralph A. Llewellyn Modern Physics 2 nd Ed. (1999

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Note: 1.It is the rest energy of any isolated system be a single atom or the entire universe that is conserved. 2.The systems rest energy may be greater than, equal to, or less than the sum of the rest energies of the constituents depending on their relative velocities and the detailed character of any interaction between them

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An important case, in which the system rest energy is less that the rest energies of its members is that of a system of particles held together by attractive forces, such as the neutrons and protons in a atomic nucleus. The rest energy of a nucleus (except that of ordinary hydrogen, which is a single proton) is less than the total rest energies of its constituent particles. The difference is called the binding energy of the nucleus. To break a nucleus up completely calls for an amount of energy at least equal to its binding energy. Arthur Beiser Concepts of Modern Physics, 6 th Ed. (2003)

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Massless particle Can a massless particle exist? The conclusion is not that massless particles necessarily occur, only that the laws of physics do not exclude the possibility as long as v=c and E=pc for them, In fact, a massless particle – the photon – indeed exists and its behaviour is as expected. Arthur Beiser Concepts of Modern Physics, 6 th Ed. (2003)

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A debate of the subject surfaced in Physics Today in 1989 when Lev Okun wrote an article urging that relativistic mass should no longer be taught. Wolfgang Rindler responded with a letter to the editors defending its continued use. In 1991 Tom Sandin wrote an article in the American Journal of Physics that argued in favour of relativistic mass.

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In a 1948 letter to Lincoln Barnett, Einstein wrote "It is not good to introduce the concept of the mass M = m/(1- v2/c2)1/2 of a body for which no clear definition can be given. It is better to introduce no other mass than `the rest mass' m. Instead of introducing M, it is better to mention the expression for the momentum and energy of a body in motion." Cited in: Lev B. Okun The Concept of Mass Physics Today. June 1989

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