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https://texercises.com/exercise/half-life/
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Exercise:
A damped oscillation has a period TO and an exponential envelope with damping coefficient deO. abcliste abc Calculate the half-life time for the exponential decrease of the amplitude. Express the result as a multiple of the oscillation period. abc What fraction of the oscillation energy is left after NO periods? abcliste

Solution:
abcliste abc T_/ ThF fracln de Th approx ThP As a multiple of the period this corresponds to fracT_/T fracThT resultnP abc The energy left after a time t is given by fracE'E_ leftfracAtA_right^ leftfracA_ e^-delta tA_right^ lefte^-delta tright^ lefte^-delta N Tright^ ratioF e^-timesdetimes NtimesT ratio approx resultratiopercP abcliste
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Exercise:
A damped oscillation has a period TO and an exponential envelope with damping coefficient deO. abcliste abc Calculate the half-life time for the exponential decrease of the amplitude. Express the result as a multiple of the oscillation period. abc What fraction of the oscillation energy is left after NO periods? abcliste

Solution:
abcliste abc T_/ ThF fracln de Th approx ThP As a multiple of the period this corresponds to fracT_/T fracThT resultnP abc The energy left after a time t is given by fracE'E_ leftfracAtA_right^ leftfracA_ e^-delta tA_right^ lefte^-delta tright^ lefte^-delta N Tright^ ratioF e^-timesdetimes NtimesT ratio approx resultratiopercP abcliste
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Damped Oscillations
Tags
energy, envelope, exponential, half-life
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Difficulty
(2, default)
Points
0 (default)
Language
ENG (English)
Type
Calculative / Quantity
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