Art_2 : Inertial Property of All Forms of Energy
- From: "GSS" <gurcharn_sandhu@xxxxxxxxx>
- Date: 11 Dec 2005 08:00:34 -0800
Inertia is an inherent property of matter and mass is the
quantitative measure of inertia. The annihilation and pair production
of some of the elementary particles has shown that the elementary
particles could be viewed as specially lumped up or locally entrapped
forms of energy. Therefore, it is quite reasonable to deduce that
Inertia must be an inherent property of all forms of energy, especially
the entrapped energy. The quantitative measure of inertia of a small
energy content dE may be given by its equivalent mass content dm
through the well known relation
dm = dE / c² ......................... (1)
where c is the speed of light in vacuum.
>>From this inertial property of all forms of entrapped energy, we can
derive the notion of dynamic mass and develop its quantitative
relationship with the rest mass. Let a material particle P be at rest
in some global inertial reference frame and let its rest mass in this
frame be m_0. When at rest, the kinetic energy of this particle P will
obviously be zero.
Now let us assume that the particle P is set in motion through
application of a constant force F. Further, at an instant of time t,
let the instantaneous velocity of P be v with corresponding kinetic
energy content E. Since the energy content E will also exhibit the
inertial property, let the quantitative measure of total inertia of P
at the instant t be given by m, which may also be referred as the
dynamic mass of the particle. If during a small interval of time dt
the particle traverses a small distance ds and gains a small amount
of kinetic energy dE then the following relations will hold.
v = ds/dt ..................... (2)
dE = F.ds ..................... (3)
>>From Newton's second law of motion
F = d(mv) / dt = m. dv/dt + v. dm/dt ............ (4)
>>From equations (3) and (4)
dE = m. dv/dt . ds + v. dm/dt . ds
= mv . dv + v². dm ............ (5)
And from equations (1) and (5) we get,
dm = (mv/c²) . dv + (v²/c²). dm ............ (6)
Let us make a substitution x= v/c in equation (6) so that dx = dv/c
and
dm = mx . dx + x² . dm ............... (7)
Or (1-x²) dm = mx . dx
Or dm/m = (x/(1-x²)) . dx ................ (8)
This on integration yields,
m/m_0 = (1-x²)^(-1/2)
Or m = m_0 / Sqrt(1- v²/c²) .................. (9)
This is a standard relation for the dynamic mass of a particle in
motion. Here it is important to note that the derivation of dynamic
mass m in terms of rest mass m_0 did not involve special
relativity. Instead this derivation is entirely based on the inertial
property of all forms of energy, including K.E. Similarly all dynamic
relations of SR can be shown to be resulting from the inertial property
of all forms of energy, including K.E.
GSS
http://groups.google.com/group/sci_physics_fundamental
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