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Saturday, September 20, 2014

What Am I Doing? Pressure Lamp

If we formulate the Lagrangian along the radial line, since/assuming all  perpendicular axes have no effects

L=TV

L=12mev2re(PEre)

PEre  is negative as zero potential is defined at  x;  in this system the forces are attractive.

we know that,

ddt{L˙re}=ddt{˙re{12mev2re+PEre}}

ddt{L˙re}=ddt{˙re{12me˙r2re+PEre}}

ddt{me˙rre+PEre˙re}

ddt{PEre˙re}=(dPEre)˙redt=(dPEredt)(dredt)=PErere

ddt{me˙rre+PEre˙re}=me¨rre+PErere

=2(PEre)re,    since  me¨re=PErere

=Lre=re{12mev2re+PEre}

If  vre=dredTdTdt,

=re{12me(dredTdTdt)2+PEre}

=me(dredTdTdt){re(dredTdTdt)}+PErere

=me(dredTdTdt){dredTre(dTdt)+dTdtre(dredT)}+PErere

=me(dredTdTdt){dredTddt(dTdre)}+PErere

=medredTdredtddt(dTdre)+PErere

So,

medredTdredtddt(dTdre)+PErere=2(PEre)re 

medredTdredtddt(dTdre)=(PEre)re

medredTd2Tdt2=(PEre)re

where dredT is negative.

This means at the kink in the  re  vs  T curve, resonance occurs at  (from the post "I like SHM, Death Rays Again, Way Cool...".),

ω2o=2(PEre)r2ekink=re{dredTd2Tdt2}=dredTkinkd2dt2{dTdrekink}

per unit mass

The temperature gradient is changing as a result of applying dTdt.  Resonance is obtained when the applied frequency is the value of the gradient at the kink point, times the second time derivative of the change in the reciprocal of the gradient at the kink point.

Since the material is radiating packets of energy,  it is expected to be damped,

We used the model,    ¨re+2p˙re+ω2ore

When the driving force is sinusoidal,

ω2n=ω2o2p2

ω2o=dredTkinkd2dt2{dTdrekink}

where  p  is the damping factor.  If  T=ToeiwatT(re)  then at maximum, d2Tdt2extrema=Tow2a  and its driving frequency = wa.   ie d2Tdt2extrema=Tow2a occurs wa times per seconds when  T is applied.

At resonance, driving frequency = damped resonance frequency,

A.Tow2a2p2=w2a

(ATo1).w2a=2p2

w2a=2p2(ATo1)

provided  ATo>1

where A=dredTkinkdTdrekink=1

So, it is possible for resonance provided,

 To>1

In a kerosene pressure lamp increasing pressure increases chemical reaction rate that in turns increases  d2Tdt2kink,  a bright glow results when resonance frequency is reached. The material in resonance glow is thorium oxide in a pressure lantern.