Pendulum with varying length, and adiabatic approximation of period; classical mechanics problem that is the springboard for adiabicity in quantum mec...
Treatment of Coriolis force, which a Foucault pendulum is designed to measure. A heavy analysis is carried out right to the end, and Taylor expansion is the very last step.Descripción completa
Physics pendulum
Full description
geometryFull description
Electronicsad communication engineeringFull description
descriptionFull description
Descripción: archivo que ayuda para el control de verticalidad de poste metalicos
Antenna ans wave propagation for 6th sem ece an important theorem in unit 1
Bandwidth TheoremFull description
This experiment was carried out to determine a value for the acceleration due to gravity g. The periods of a simple pendulum were timed and recorded for each of the ten variations in the len…Full description
nmn
Bayes Theorem
tFull description
experiment sheet for compound pendulumFull description
Pendulum
trhyjyyj
Full description
ScrollsawingDescripción completa
Simple example of Chinese Remainder TheoremFull description
bradley j. nartowt Saturday, July 06, 2013, 09:22:21
PHYS 6246 – classical mechanics Dr. Whiting Whiting
By considering considering the work work done to alter adiabaticall adiabaticallyy the length length of a plane pendulum, prove by elementary means the adiabatic invariance of J for the plane pendulum in the limit of vanishing amplitude. Draw a picture of this pendulum,
[I.1] Action variable for a pendulum whose string is not being not being toyed with,
J
p dq
2m
2
E 0 mg cos d E 0
0
/ g
[I.2]
Also: total energy of pendulum is,
E0 12 m( g / )(
max )
2
12 m(
max )
2
[I.3]
Elementary means: just consider a freshman physics force balance. Find force in y direction only,
F F
g
Fshorten length m s mgs m ˆ
ˆ
2
y Fy mg cos m ˆ
2
[I.4]
Taking the time-average of this force, and using the vanishing amplitude cos 1 12 O( ) , we get, 2
F
mg 1 12 2 m
2
mg 12 mg
mg mgmax m( g / )( 1 4
2
1 2
max )
2
2
m
2
mg 12 mg
4
2 1 2 max
m
2 1 2 max
mg E0 / 2 E0 / mg E0 / 2
[I.5]
Do a work- balance: balance: work done on pendulum to adiabatically adiabatically change change string string length is “endothermic” “endothermic” (universe (universe loses loses energy, so affix a minus sign). Then, separate variables and integrate,
d ( E0 mg ) dW Looking at [I.2], we solve for the quantity E0
ln E0 ln
/g
f
E dy mg 0 d 2
dE 0
ˆ
E0
d 2( / g )
[I.6]
/ g , and see that it is constant, meaning its total derivative is zero,
