Name: period and frequency in simple harmonic motion
Uploaded: 2023-05-16T08:18:40.000Z
Duration: 1 min 21 s
Channel: JoVE

K12 Standards

ohios-learning-standards-and-model-curriculum

Ohio’s Learning Standards and Model Curriculum

9-12

Physics

Forces, Momentum And Motion

Elastic forces

Simple Harmonic Motion

Wenn sich bewegende Objekte ihren Weg über einen regelmäßigen Zeitraum wiederholen, spricht man von harmonischer oder periodischer Bewegung. Die periodische Bewegung, bei der sich ein Objekt von einem festen Punkt hin und her bewegt, wird als Oszillation bezeichnet.
Wenn die Rückstellkraft auf das Objekt proportional zur Verschiebung aus der Gleichgewichtsposition ist, wird die Schwingung als einfache harmonische Bewegung bezeichnet. Allerdings sind nicht alle periodischen Schwingungen einfache harmonische Bewegungen.
Stellen Sie sich ein Lineal vor, bei dem ein Ende an der Tischplatte befestigt ist und das andere frei mit einer angehängten Masse versehen ist. Wenn dieses System hochgezogen und losgelassen wird, vibriert es.
Dabei ist die Beschleunigung auf das Schwingungszentrum gerichtet und proportional zur Verschiebung aus der Gleichgewichtsposition. Dadurch ist die Kraft proportional zur Verschiebung, aber in die entgegengesetzte Richtung. Die Proportionalitätskonstante ist die Federkonstante.
Eine vollständige Schwingung wird als Zyklus bezeichnet, und die Zeit, die benötigt wird, um einen Zyklus abzuschließen, wird als Periode bezeichnet. Die Anzahl der Zyklen pro Zeiteinheit ist die Häufigkeit.
Im Gleichgewicht ist die Nettokraft Null. Die maximale Verschiebung aus der Gleichgewichtsposition ist die Amplitude.

<ol>
	<li>OpenStax. (2019). Universitätsphysik Vol. 1. [Webversion]. Abgerufen von (Seite Nr. 744-745) <a href="https://openstax.org/books/university-physics-volume-1/pages/1-introduction">https://openstax.org/books/university-physics-volume-1/pages/1-introduction</a></li>
	<li>OpenStax. (2020). College Physik [Webversion]. (S. Nr. 671-674) <a href="https://openstax.org/details/books/college-physics">https://openstax.org/details/books/college-physics</a></li>
</ol>

einfache harmonische bewegung

Einfache harmonische Bewegung ist die Bezeichnung für die oszillatorische Bewegung eines Systems, bei dem die resultierende Kraft durch das Hookesche Gesetz beschrieben werden kann. Wenn die resultierende Kraft durch das Hookesche Gesetz beschrieben werden kann und keine Dämpfung (durch Reibung oder andere nicht-konservative Kräfte) vorliegt, dann wird ein einfacher harmonischer Oszillator mit gleicher Auslenkung auf beiden Seiten der Gleichgewichtslage oszillieren. Um eine Gleichung für die Periode und die Frequenz abzuleiten, wird die Bewegungsgleichung verwendet. Die Periode eines einfachen harmonischen Oszillators wird durch die folgende Gleichung gegeben:

<img alt="Equation1" src="/files/ftp_upload/12762/12762_Equation_1v2.png" style="text-align: center; height: 35px;" />

Und da Frequenz und Periode eine umgekehrte Beziehung haben, ist die Frequenz eines einfachen harmonischen Oszillators:

<img alt="Equation2" src="/files/ftp_upload/12762/12762_Equation_2v2.png" style="text-align: center; height: 35px;" />

Beachten Sie, dass weder die Periode noch die Frequenz von der Amplitude abhängen und die SI-Einheit für Frequenz Hertz ist.

Betrachten wir nun jeweils ein Beispiel. (a) Ein medizinisches Bildgebungsgerät erzeugt Ultraschall durch eine Oszillation mit einer Periode von 0,400 μs. Welche Frequenz hat diese Schwingung? (b) Die Frequenz des mittleren C auf einem typischen Musikinstrument beträgt 264 Hz. Wie lange dauert eine vollständige Schwingung?

Beide können mithilfe der inversen Beziehung zwischen Periode und Frequenz beantwortet werden. Durch Einsetzen des gegebenen Wertes der Periode in den Frequenzausdruck wird die Frequenz der Schwingung gefunden:

<img alt="Equation3" src="/files/ftp_upload/12762/12762_Equation_3v2.png" style="text-align: center; height: 50px;" />

Durch Einsetzen des gegebenen Wertes für die Frequenz in den Ausdruck für die Zeitperiode, erhält man die Zeit für eine vollständige Schwingung:

<img alt="Equation4" src="/files/ftp_upload/12762/12762_Equation_4v2.png" style="text-align: center; height: 50px;" />

Einfache harmonische Bewegung

Einfache harmonische Bewegung ist die Bezeichnung für eine oszillierende Bewegung für ein System, bei dem die Nettokraft durch das Hookesche Gesetz beschrieben werden kann. Wenn die Nettokraft ...

/files/media/science-education/science-education-thumbs/12762.jpg

Education

Lehre

JoVE Core

Physik

Unit 1

Einheit 1

Oscillations

Schwingungen

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

period and frequency in simple harmonic motion

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 be converted from one to the other. The SI unit for frequency is Hertz.

Neither period nor frequency depends on amplitude. That means the size of the swing does not change these values in simple harmonic motion. This idea helps when solving problems about oscillation time and rate.

The inverse relationship can be used in real examples. A medical imaging device that produces ultrasound may oscillate with a period of 0.400 μs, and its frequency can be found from that value. Middle C on a typical musical instrument has a frequency of 264 Hz, and the time for one complete oscillation can be found from the same relationship.

Period and Frequency in Simple Harmonic Motion

period-and-frequency-in-simple-harmonic-motion

simple harmonic motion

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...

characteristics of simple harmonic motion

characteristics-of-simple-harmonic-motion

Simple Harmonic Motion: Period and Phase

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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oscillations about an equilibrium position

oscillations-about-an-equilibrium-position

Stable Equilibrium and Atomic Oscillation

Stable equilibrium is a key idea in oscillation. When an object sits at a stable equilibrium point, a small push moves it away, but it then tends to oscillate around that point. If the equilibrium point is unstable, a slight disturbance will not bring the object back. The transcript also describes a half-stable equilibrium point. It is still considered unstable, but the name comes from the fact that the force on only one side points toward the equilibrium point. That makes it different from a...

energy in simple harmonic motion

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.

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frequency of spring-mass system

How Mass and Spring Stiffness Set Oscillation

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 and uniform circular motion

simple-harmonic-motion-and-uniform-circular-motion

Simple Harmonic Motion from Circular Motion

Simple harmonic motion and uniform circular motion are closely linked. Simple harmonic motion is oscillatory motion in a system where the net force follows Hooke’s law. Uniform circular motion is motion along a circular path at constant speed. A simple way to create simple harmonic motion is to use uniform circular motion. One example is a ball attached to a vertical turntable that spins at a steady rate. If you watch the ball’s shadow on the floor, the shadow, or projection, moves in simple...

problem solving: energy in simple harmonic motion

problem-solving-energy-in-simple-harmonic-motion

Energy Conservation in Simple Harmonic Motion

Energy conservation in simple harmonic motion explains how a spring and mass system shares energy as it moves back and forth around equilibrium. Simple harmonic motion, or SHM, is periodic motion in time and position. The object oscillates with a constant amplitude and frequency. In SHM, kinetic energy and potential energy change into each other continuously. A good example is the spring in a car shock absorber as the car moves on a bumpy road. The wheel attached to the spring oscillates in...

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simple pendulum

Pendulum Motion and Period

A simple pendulum is a small ball, or bob, hanging from a string that has negligible mass and does not stretch. Pendulums appear in everyday life, including clocks, swing sets, and a sinker on a fishing line. Their motion is a useful model for studying repeating movement. The period of a simple pendulum depends on two factors: its length and the acceleration due to gravity. The period does not depend on the bob’s mass or on the maximum displacement. For small angles, the pendulum behaves like...

torsional pendulum

Angular Oscillation in a Torsion String

Angular oscillation in a torsion string is shown by a rigid body suspended from a string that twists as it moves. The twisting of the string provides the restoring force. The string is ideally massless, and in practice its mass is much smaller than the rigid body&amp;#39;s mass, so it is neglected. When the angular displacement is small, the motion can be modeled as linear angular oscillation. The amplitude is measured as an angle. In this system, the role of mass is played by the rigid...

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physical pendulum

Physical Pendulum Motion and Inertia

A physical pendulum is a rigid body that hangs from a fixed pivot point and swings when it is displaced. Its motion is similar to a simple pendulum. To describe that motion, physicists use the small-angle approximation, where sin θ ≈ θ. This approximation is valid up to about 14°. The period and angular frequency of a physical pendulum can be found by comparing it with a spring-mass system. The mass moment of inertia is the key quantity that describes how the body’s mass is spread around the...

/files/media/science-education/science-education-thumbs/14719.jpg

measuring acceleration due to gravity

Using Pendulums to Find Gravity

Pendulums can be used to measure the acceleration due to gravity. A simple pendulum is a point mass attached to a string. A physical pendulum is an object that swings in a similar way, but its mass is spread out, so it cannot be treated as a point mass on a string. A physical pendulum is described using rotational motion and moment of inertia. In both types of pendulum, gravity provides the restoring force that brings the object back toward the middle of its swing. For a simple pendulum,...

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damped oscillations

Energy Loss in Oscillations

Oscillations in the real world often lose energy as they move. A system that keeps the same amplitude without any loss is called undamped. That kind of motion is rare, because friction and other non-conservative forces usually reduce the motion over time. Common examples show this clearly. A guitar string stops vibrating a few seconds after it is plucked. A swing also needs a person to keep pushing it if the motion is to continue. In some devices, such as car shock absorbers, damping is useful...

types of damping

How Damping Changes Oscillations

Damping changes how a system oscillates by reducing its back-and-forth motion. As damping increases, the period and frequency begin to change because damping opposes motion and lowers the net force in both directions. When damping is very large, the system stops oscillating altogether. It then moves slowly toward equilibrium. An overdamped system reaches equilibrium slowly. An underdamped system reaches equilibrium faster, but it oscillates around the equilibrium point while doing so. A...

forced oscillations

Resonance in Driven Oscillators

Resonance in driven oscillators describes how a periodic force changes an object’s motion. A driven oscillator may first move in a messy way. These early motions are called transients. After the transients die out, the oscillator reaches a steady state. In this state, the motion becomes periodic and the displacement is determined. The amplitude of a driven oscillator can be found from the equations for force and motion. That amplitude depends on the natural frequency of the system and the...

concept of resonance and its characteristics

concept-of-resonance-and-its-characteristics

Resonance in Music, Cars, and Radios

Resonance is a key feature of driven oscillators, and it depends strongly on damping. Damping is the process that reduces motion by removing energy from the system. When a system must resonate at one specific frequency, very light damping is needed. Piano strings and many other musical instruments are common examples of this kind of response. Heavy damping works better when the goal is small-amplitude oscillations. A car suspension system uses this approach so that bumps do not produce large...

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key terms and concepts

Learn how period and frequency are related in simple harmonic motion, why amplitude does not matter, and how to solve oscillation examples in Hertz.

Period and Frequency in Simple Harmonic Motion | JoVE K12

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Tennessee Academic Standards

High School

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Waves and Their Applications in Technologies for Information Transfer

PHYS.PS4.2

Florida Department of Education

Physical Science

SC.912.P.10

SC.912.P.10.6

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PHYS.8

PHYS.8.A

Chemistry and Physics

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Newton's laws applied to complex problems

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Gravitational force and fields

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Gravitational force and fields

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Characteristics of Simple Harmonic Motion

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Hooke's Law

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Stress

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Strain and Elastic Modulus

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Elasticity

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Elastic forces

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Friction force (static and kinetic)

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Momentum, impulse and conservation of momentum

Period and Frequency in Simple Harmonic Motion

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