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Eigenfrequenz einer Feder

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Exercise

https://texercises.com/exercise/eigenfrequenz-einer-feder/

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Exercise:
An einer Feder der Masse m_F schwingt die Masse m. Berechne die Eigenfrequenz.

Solution:
Die gesamte kinetische Energie besteht aus jener des Körpers am Ende der Feder sowie jener der Feder. Dabei ist jedoch zu beachten dass sich nicht jeder infinitesimal kleine Teil der Feder gleich schnell bewegt. Viel mehr nimmt die Geschwindigkeit dieser Teilchen linear zu und zwar gilt dot xi fracxil dot x. Dabei ist dot x die Geschwindigkeit der Masse am Ende und dot xi die Geschwindigkeit des Federteilchens an der Stelle xi. Die Federlänge ist mit l bezeichnet. Damit kann die kinetische Energie wie folgt geschrieben werden: Ekin Ekin^M + Ekin^F frac mdot x^ + frac _^L mudotxi^ mboxdxi fracdot x leftm+fracmu Lright fracdot x leftm+fracm_Fright. Aufgrund des Energieerhaltungssatzes gilt nun dass die kinetische Energie gleich der potentiellen der Feder ist: Ekin frac fx^ Es gilt nun: fracpartialpartial t Ekin fracpartialpartial t frac fx^ leftm+fracm_Fright ddot x + fx Daraus kann die Eigenfrequenz direkt abgelesen werden: f fracpi sqrtfracfm+fracm_F

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Contained in these collections:

ETH 1. Vordiplom Physik Herbst 1993 by TeXercises

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Attributes & Decorations

Tags	eigenfrequenz, eigenmasse, eigenschwingung, feder, federgesetz, frequenz, harmonische, hookesches gesetz, mechanik, oszillator, physik, schwingung
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Difficulty	(5, default)
Points	5 (default)
Language	(Deutsch)
Type	Calculative / Quantity
Creator	uz
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