L=T−V
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)=∂PEre∂re
ddt{me˙rre+∂PEre∂˙re}=me¨rre+∂PEre∂re
=2∂(PEre)∂re, since me¨re=∂PEre∂re
=∂L∂re=∂∂re{12mev2re+PEre}
If vre=dredTdTdt,
=∂∂re{12me(dredTdTdt)2+PEre}
=me(dredTdTdt){∂∂re(dredTdTdt)}+∂PEre∂re
=me(dredTdTdt){dredT∂∂re(dTdt)+dTdt∂∂re(dredT)}+∂PEre∂re
=me(dredTdTdt){dredTddt(dTdre)}+∂PEre∂re
=medredTdredtddt(dTdre)+∂PEre∂re
So,
medredTdredtddt(dTdre)+∂PEre∂re=2∂(PEre)∂re
medredTdredtddt(dTdre)=∂(PEre)∂re
medredTd2Tdt2=∂(PEre)∂re
where dredT is negative.
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.
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,
We used the model, ¨re+2p˙re+ω2ore
When the driving force is sinusoidal,
ω2n=ω2o−2p2
ω2o=−dredTkinkd2dt2{dTdrekink}
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.Tow2a−2p2=w2a
(ATo−1).w2a=2p2
w2a=2p2(ATo−1)
provided ATo>1
where A=dredTkinkdTdrekink=1
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.