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

Рассмотрим блок массы m, прикрепленный к пружине с постоянной силой k на поверхности без трения в равновесии.
При смещении блока работа, выполняемая усилием пружины при перемещении из начального положения в конечное, равна количеству потенциальной энергии, накопленной в пружине, относительно ее перемещения. Это называется упругой потенциальной энергией.
При отпускании блок претерпевает простое гармоническое движение. Энергия, необходимая для его движения вперед-назад, называется поступательной кинетической энергией и прямо пропорциональна квадрату его скорости и массы.
Во время колебаний обе энергии непрерывно меняются и представлены синусоидальными формами волн. Потенциальная энергия упругости максимальна при максимальном смещении, в то время как поступательная кинетическая энергия максимальна в положении равновесия.
В других положениях блок имеет разные значения кинетической и потенциальной энергии, и их сумма равна суммарной энергии системы.
Таким образом, общая энергия в системе остается постоянной и сохраняется, поскольку она колеблется между поступательной кинетической энергией и упругой потенциальной энергией.

<ol>
	<li>OpenStax. (2020). Физика колледжа [Веб-версия]. (Стр. No 680-682) <a href="https://openstax.org/details/books/college-physics">https://openstax.org/details/books/college-physics</a></li>
</ol>

Энергия в простом гармоническом движении

Чтобы определить энергию гармонического осциллятора, рассмотрим все формы энергии, которые он может иметь во время своего простого гармонического движения. Согласно закону Гука, энергия, запасённая при сжатии/растяжении струны в простом гармоническом осцилляторе, является потенциальной энергией. Так как простой гармонический осциллятор не имеет диссипативных сил, он также обладает кинетической энергией. При наличии консервативных сил обе энергии могут относиться друг к другу во время колебаний, но общая энергия остаётся постоянной. Общая энергия простого гармонического осциллятора равна сумме потенциальной и кинетической энергии и пропорциональна квадрату амплитуды. Её можно выразить в следующей форме:

<img alt="Equation1" src="/files/ftp_upload/12765/12765_Equation_1.svg" style="text-align: center; height: 35px;" />

Величина скорости в простом гармоническом движении получается путем перестановки и решения уравнений общей энергии.

<img alt="Equation2" src="/files/ftp_upload/12765/12765_Equation_2.svg" style="text-align: center; height: 42px;" />

Манипулируя этим алгебраическим выражением, получаем следующее:

<img alt="Equation3" src="/files/ftp_upload/12765/12765_Equation_3.svg" style="text-align: center; height: 50px;" />

где

<img alt="Equation4" src="/files/ftp_upload/12765/12765_Equation_4.svg" style="text-align: center; height: 42px;" />

Заметьте, что максимальная скорость зависит от трёх факторов и пропорциональна амплитуде. Если смещение максимально, то и скорость также будет максимальной. Кроме того, максимальная скорость больше для более жёстких систем, поскольку они оказывают большую силу при том же смещении. Это наблюдение можно увидеть в выражении для максимальной скорости. Максимальная скорость пропорциональна квадратному корню из постоянной силы. Наконец, максимальная скорость меньше для объектов с большой массой, поскольку максимальная скорость обратно пропорциональна квадратному корню из массы.

Чтобы определить энергию простого гармонического осциллятора, рассмотрим все формы энергии, которые он может иметь во время своего простого гармонического движения. Согласно...

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

Education

JoVE Core

Physics

Физика

Unit 1

Раздел 1

Oscillations

Колебания

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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

simple harmonic motion

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...

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characteristics of simple harmonic motion

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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

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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 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.

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

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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

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physical pendulum

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measuring acceleration due to gravity

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damped oscillations

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forced oscillations

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concept of resonance and its characteristics

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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 simple harmonic motion stores potential and kinetic energy, why total energy stays constant, and what controls maximum velocity.

Simple Harmonic Motion: Energy and Speed | JoVE K12

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3.2.9-12.Q

Louisiana Student Standards for Science

High School

HS-PS3-2

HS-PS3-3

Tennessee Academic Standards

waves-and-their-applications-in-technologies-for-information-transfer

Waves and Their Applications in Technologies for Information Transfer

PHYS.PS4.2

Next Generation Science Standards

HS-PS3-1

HS-PS3-5

Florida Department of Education

SC.912.P.10

SC.912.P.10.6

Texas Essential Knowledge and Skills

PHYS.8

PHYS.8.A

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3.2.9-12.O

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3.2.9-12.R

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Energy Diagrams - II

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3.2.9-12.S

Simple Harmonic Motion: Energy and Speed

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