Name: simple harmonic motion: energy and speed
Uploaded: 2025-05-22T09:05:41.000Z
Duration: 1 min 23 s
Channel: JoVE
Description: Simple harmonic motion combines potential energy and kinetic energy. In a spring-like oscillator, Hooke’s law explains the potential energy stored when the string or spring is stretched or compressed. Because there are no dissipative forces, the object also has kinetic energy as it moves.

K12 Standards

Pennsylvania State Academic Standards

9-12

Physical Science

Energy

3.2.9-12.P

Energy in Simple Harmonic Motion

Betrachten Sie einen Massenblock m, der mit einer konstanten Kraft k auf einer reibungsfreien Oberfläche im Gleichgewicht an einer Feder befestigt ist.
Wenn der Block verschoben wird, entspricht die Arbeit, die die Federkraft entlang der Verschiebung von der Anfangs- zur Endposition leistet, der Menge an potentieller Energie, die in der Feder relativ zu ihrer Verschiebung gespeichert ist. Dies wird als elastische potentielle Energie bezeichnet.
Beim Loslassen erfährt der Block eine einfache harmonische Bewegung. Die Energie, die für seine Hin- und Herbewegung erforderlich ist, wird als translationale kinetische Energie bezeichnet und ist direkt proportional zum Quadrat seiner Geschwindigkeit und seiner Masse.
Während der Schwingungen tauschen sich beide Energien kontinuierlich aus und werden durch sinusförmige Wellenformen dargestellt. Die elastische potentielle Energie ist bei maximaler Verschiebung am Maximum, während die translationale kinetische Energie bei der Gleichgewichtsposition am Maximum ist.
An anderen Positionen hat der Block unterschiedliche kinetische und potentielle Energiewerte, deren Summe der Gesamtenergie des Systems entspricht.
So bleibt die Gesamtenergie im System konstant und erhalten, da sie zwischen translationaler kinetischer Energie und elastischer potentieller Energie oszilliert.

<ol>
	<li>OpenStax. (2020). College Physik [Webversion]. (Seite Nr. 680-682) <a href="https://openstax.org/details/books/college-physics">https://openstax.org/details/books/college-physics</a></li>
</ol>

energie in harmonischer bewegung

Um die Energie eines harmonischen Oszillators zu bestimmen, betrachten wir alle Formen der Energie, die er während seiner harmonischen Bewegung haben kann. Gemäß dem Hookschen Gesetz ist die Energie, die während der Kompression/Dehnung eines Seils in einem harmonischen Oszillator gespeichert ist, potenzielle Energie. Da der harmonische Oszillator keine dissipativen Kräfte besitzt, hat er auch kinetische Energie. In Bezug auf konservative Kräfte können sich beide Energien während der Schwingung ineinander umwandeln, aber die Gesamtenergie bleibt konstant. Die Gesamtenergie für einen harmonischen Oszillator ist gleich der Summe aus potenzieller und kinetischer Energie und ist proportional zum Quadrat der Amplitude. Sie kann in folgender Form ausgedrückt werden:

<img alt="Equation1" src="/files/ftp_upload/12765/12765_Equation_1.svg" style="text-align: center; height: 35px;" />

Die Größe der Geschwindigkeit in einer harmonischen Bewegung wird durch Umstellen und Lösen der Gleichungen für die Gesamtenergie erhalten.

<img alt="Equation2" src="/files/ftp_upload/12765/12765_Equation_2.svg" style="text-align: center; height: 42px;" />

Durch algebraische Manipulation dieses Ausdrucks ergibt sich folgendes:

<img alt="Equation3" src="/files/ftp_upload/12765/12765_Equation_3.svg" style="text-align: center; height: 50px;" />

wobei

<img alt="Equation4" src="/files/ftp_upload/12765/12765_Equation_4.svg" style="text-align: center; height: 42px;" />

Beachten Sie, dass die maximale Geschwindigkeit von drei Faktoren abhängt und proportional zur Amplitude ist. Wenn die Verschiebung maximal ist, ist auch die Geschwindigkeit maximal. Zusätzlich ist die maximale Geschwindigkeit bei steiferen Systemen größer, da sie für die gleiche Verschiebung eine größere Kraft ausüben. Diese Beobachtung findet sich im Ausdruck für die maximale Geschwindigkeit wieder. Die maximale Geschwindigkeit ist proportional zur Quadratwurzel der Federkonstante. Schließlich ist die maximale Geschwindigkeit bei Objekten mit größerer Masse kleiner, da die maximale Geschwindigkeit umgekehrt proportional zur Quadratwurzel der Masse ist.

Energie in harmonischer Bewegung

Um die Energie eines einfachen harmonischen Oszillators zu bestimmen, betrachten Sie alle Energieformen, die er während seiner einfachen harmonischen Bewegung haben kann. Laut...

/files/media/science-education/science-education-thumbs/12765.jpg

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simple harmonic motion: energy and speed

Simple harmonic motion combines potential energy and kinetic energy. In a spring-like oscillator, Hooke’s law explains the potential energy stored when the string or spring is stretched or compressed. Because there are no dissipative forces, the object also has kinetic energy as it moves.

Simple Harmonic Motion: Energy and Speed

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Simple harmonic motion describes oscillatory motion when the net force follows Hooke's law. In this type of motion, the force acts toward the equilibrium position. If there is no damping from friction or other non-conservative forces, the oscillator moves with equal displacement on both sides of equilibrium. The period of a simple harmonic oscillator is found from the equation of motion. Period is the time for one complete oscillation. Frequency is the inverse of period, so the two values can...

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Simple harmonic motion depends on how displacement, force, and acceleration work together. In this motion, the acceleration of the system, and therefore the net force, is proportional to the displacement and points in the opposite direction. That restoring force pulls the object back toward equilibrium. The period and frequency of a simple harmonic oscillator do not depend on amplitude. A diving board shows this idea well. A stiff, thick diving board has a large force constant, so it has a...

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energy in simple harmonic motion

Simple harmonic motion combines potential energy and kinetic energy. In a spring-like oscillator, Hooke’s law explains the potential energy stored when the string or spring is stretched or compressed. Because there are no dissipative forces, the object also has kinetic energy as it moves.

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A spring-mass system shows simple harmonic motion, and its oscillation rate depends on the mass and the spring force constant. In this model, the angular frequency, period, and frequency are set by those two values. They do not depend on the motion’s amplitude or on the initial conditions. To find these relationships, the system can be analyzed with Newton’s second law. Imagine a block on a spring on a frictionless surface. Three forces act on the mass: weight, the normal force, and the spring...

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Simple Harmonic Motion from Circular Motion

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Simple Harmonic Motion: Energy and Speed | JoVE K12

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3.2.9-12.O

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3.2.9-12.R

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3.2.9-12.S

Simple Harmonic Motion: Energy and Speed

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