Name: energy conservation in simple harmonic motion
Uploaded: 2023-05-16T08:18:41.000Z
Duration: 1 min 17 s
Channel: JoVE

K12 Standards

Pennsylvania State Academic Standards

9-12

Physical Science

Energy

3.2.9-12.P

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

Problem Solving: Energy in Simple Harmonic Motion

でこぼこの道を車が動いているとします。ここでは、車のサスペンションは、車の質量を車輪に接続するバネのような役割を果たします。
その動作中、スプリングは圧縮および膨張し、単純な調和運動を実行します。
車がある振幅で跳ね返ると仮定します。車の最大速度はどれくらいで、初期位置と平衡位置の中間にある場合、その全エネルギーはどのように変化するのでしょうか。
単純調和運動の総エネルギーは保存されたままです。項の再配置により、モーション中の任意の位置での速度が得られます。
最大速度は、車のスプリングが平衡位置を通過するときに発生します。既知の量を代入すると、最大速度が得られます。次に、平衡位置での全エネルギーは、既知で計算された量を代入することによって得られます。
車のスプリングが最大振幅から半分の振幅に移動すると、速度は負と見なされます。総エネルギーは、速度を他の既知の量に代入することによって計算されます。
どちらの場合も、単純調和運動の総エネルギーは同じです。

<ol>
	<li>ヤング、H.D.およびフリードマン、RA(2012)。現代物理学と大学物理学。カリフォルニア州サンフランシスコ:ピアソン。446ページ。</li>
	<li>OpenStaxです。(2019). 大学物理学 Vol.1.【Web版】。754ページ。<a href="https://openstax.org/books/university-physics-volume-1/pages/15-2-energy-in-simple-harmonic-motion"から取得>https://openstax.org/books/university-physics-volume-1/pages/15-2-energy-in-simple-harmonic-motion</a></li>
</ol>

問題解決: 単振動のエネルギー

単調和運動 (SHM) は、時間と位置の周期運動の一種であり、物体が一定の振幅と周波数で平衡位置の周りを前後に振動します。 SHM では、位置エネルギーと運動エネルギーの間で継続的な交換が行われ、その結果、物体が振動します。

車のショックアブソーバーのバネを考えてみましょう。 でこぼこした路面を走行するとき、車輪に取り付けられたばねは単振動を行います。 ばねにかかる力は保守的であり、ばねが伸張または圧縮されるときに位置エネルギーが蓄えられます。 この場合、バネに取り付けられたホイールは 1 次元で振動し、バネの力が動きと平行に作用します。 平衡位置では、ばねに蓄えられる位置エネルギーはゼロになります。 散逸力がない場合、総エネルギーは位置エネルギーと運動エネルギーの合計であり、次のように表されます。

<img alt="Equation1" src="/files/ftp_upload/14743/14743_Equation_1.svg" style="text-align: center; height: 35px;" />

単振動の総エネルギーは、運動中のすべての点で系に保存され、振幅の 2 乗に比例します。

単振動の全エネルギー方程式は、速度、位置、および全機械エネルギーの間の有用な関係を示します。 この方程式は、問題で時間を考慮せずに位置、速度、加速度の関係を必要とする場合に使用できます。 エネルギー保存方程式には変位と速度が含まれるため、状況から変位と速度の符号を推測する必要があります。 たとえば、物体が平衡位置から最大の正の変位点に向かって移動する場合、変位と速度は正の値になります。

単振動のエネルギーを研究することは、物理学や工学における振動系の動作を理解するために不可欠です。

単純調和運動(SHM)は、時間と位置における周期運動の一種であり、物体が平衡の周りで前後に振動します...

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Education

JoVE Core

Physics

物理学

Unit 1

ユニット1

Oscillations

振動

KEY TERMS AND CONCEPTS

主要用語と概念

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 one dimension, and the spring force acts parallel to the motion. The force is conservative, so energy is stored as potential energy when the spring is stretched or compressed.

At the equilibrium position, the spring’s potential energy is zero. If no dissipative forces are present, the total mechanical energy stays constant throughout the motion. This total energy is the sum of kinetic energy and potential energy. In SHM, it is also proportional to the square of the amplitude.

The total energy equation in SHM links position, velocity, and mechanical energy. It is useful when a problem asks for a connection between position, velocity, and acceleration without using time. Because the equation includes displacement and velocity, the signs must be inferred from the motion. For example, if the object moves from equilibrium toward the greatest positive displacement, both displacement and velocity are positive.

Understanding energy in simple harmonic motion helps explain how oscillating systems behave in physics and engineering.

Energy Conservation in Simple Harmonic Motion

energy-conservation-in-simple-harmonic-motion

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

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...

/files/media/science-education/science-education-thumbs/12763.jpg

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...

/files/media/science-education/science-education-thumbs/14718.jpg

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

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...

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...

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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 kinetic and potential energy change in simple harmonic motion, with a spring example and the energy equation used to solve SHM problems.

Energy Conservation in Simple Harmonic Motion | JoVE K12

physics

JoVE Core 物理学は、運動学、力学、流体力学、静力学、波動、音響などの基礎的な物理学の理論と応用を、簡潔でわかりやすいアニメーション ビデオで紹介します。さらに、科学者の実際のビデオでは、世界中の今日の研究所で行われている古典的および独自の研究実験における関連概念の適用を示しています。

3.2.9-12.Q

Louisiana Student Standards for Science

High School

HS-PS3-2

HS-PS3-3

Next Generation Science Standards

HS-PS3-1

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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Energy in Simple Harmonic Motion

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3.2.9-12.R

force-and-potential-energy-in-one-dimension

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force-and-potential-energy-in-three-dimensions

Force and Potential Energy in Three Dimensions

Consider a particle moving under the action of a conservative force that has components along each coordinate axis. Each component of force is a function ...

Energy Diagrams - I

The dynamics of a mechanical system can be easily understood by interpreting a potential energy diagram. Since energy is a scalar quantity, the ...

Energy Diagrams - II

Energy diagrams are important to understand the dynamics of a system. The topology of an energy diagram helps illustrate the equilibrium points of the ...

Students who demonstrate understanding can develop and use a model of two objects interacting through electric or magnetic fields to illustrate the forces between objects and the changes in energy of the objects due to the interaction. Clarifying Statement: Examples of models could include drawings, diagrams, and texts, such as drawings of what happens when two charges of opposite polarity are near each other. Assessment Boundry: Assessment is limited to systems containing two objects.

3.2.9-12.S

Energy Conservation in Simple Harmonic Motion

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