Name: simple harmonic motion from circular motion
Uploaded: 2023-04-30T11:08:32.000Z
Duration: 1 min 42 s
Channel: JoVE

K12 Standards

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North Carolina K-12 Science Standard Course of Study

9-12

Earth and Environmental Science

Earth's Place in the Universe

ESS.EES.1.2

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

Simple Harmonic Motion and Uniform Circular Motion

地球の中心からの距離Aにある月が、一定の角速度で円運動で回転していると考えてください。
地球の中心を変位-時間座標系の原点とします。月が位置Pに移動すると、x軸上の投影P'は角度Фになります。
月が地球の周りをいつでもtの周りを移動すると、角度ωt+Фになります。x軸またはy軸上の月の投影に基づいて、投影の位置はcosine関数またはsine関数のいずれかで表すことができます。
月の周期は、地球の軌道の円周とその速度で表すことができます。エネルギー保存から速度方程式を呼び戻し、それを修正すると、月の投影の周期が決定されます。
月の速度は接線方向に作用しますが、月の加速度は半径方向の内側に向けられます。
月の速度と加速度のx成分は、月の投影の速度と加速度に等しくなります。それらの大きさは、速度と加速度の方程式を思い出すことによって得られます。
観察されたように、月の投影の期間、位置、速度、および加速度の方程式は、単純な調和振動子の方程式と似ています。
したがって、円運動が発生する円の直径に沿って均一な円運動を投影することは、単純な調和運動を表します。

単振動と等速円運動

単純な調和運動と等速円運動は 2 つの別個の概念ですが、それらは相互に相関し、相互に関連しています。 単調和運動は、正味の力がフックの法則で記述できる系内の振動運動です。一方、等速円運動は、一定の速度で円軌道を描く物体の運動です。

等速円運動を使用して、単純な調和運動を生成する簡単な方法があります。 たとえば、均一に回転する垂直ターンテーブルにボールが取り付けられており、その影が床に投影されているとします。 ここで、影の位置 (投影とも呼ばれます) は単振動を実行します。 フックの法則は通常、目に見える大きな変位を持つ系ではなく、均一な円運動 (⍵定数) を記述します。 したがって、等速円運動の投影を観察することは、多くの場合、正確で大規模な単調和振動子を観察するよりも簡単です。

もう 1 つの例は、等速円運動をするレコード プレーヤーです。 ターンテーブルの外縁の一箇所にダボ棒が取り付けられ、ダボのもう一方の端にペンが取り付けられていると考えてください。 レコードプレーヤーが回転するとペンも動きます。 長い紙をペンの下にドラッグすると、その動きが波として捉えられます。 このことから、単調和運動は、円運動が発生する円の直径に沿った等速円運動の投影であることが理解できます。

このテキストは、<a href="https://openstax.org/books/college-physics/pages/16-6-uniform-circular-motion-and-simple-harmonic-motion"> から編集されたものです。 Openstax, College Physics, Section 16.6: Uniform Circular Motion and  Simple Harmonic Motion</a><a href="https://openstax.org/books/university-physics-volume-1/pages/15-3-comparing- simple-harmonic-motion-and-circular-motion">Openstax, University Physics Volume 1, Section 15.3: Comparing Simple Harmonic Motion and Circular Motion.</a>

単純な調和運動と一様円運動は2つの別々の概念であるかもしれませんが、それらは互いに相関し、相互に関連しています。シンプルハーモニック...

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

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 harmonic motion.

This link also helps explain why circular motion can be easier to study in some cases. Hooke’s law usually describes uniform circular motions with constant angular speed, rather than systems with large visible displacements. For that reason, the projection of uniform circular motion is often easier to observe than a precise large-scale simple harmonic oscillator.

A record player gives another clear example. A dowel rod is attached to one point on the outside edge of a turntable, and a pen is attached to the other end of the dowel. As the record player turns, the pen moves. If a long piece of paper is pulled under the pen, the motion is traced as a wave. This shows that simple harmonic motion can be seen as the projection of uniform circular motion along the diameter of the circle.

Simple Harmonic Motion from Circular Motion

simple-harmonic-motion-from-circular-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...

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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 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...

/files/media/science-education/science-education-thumbs/14743.jpg

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

Explore how uniform circular motion creates simple harmonic motion through projections, shadows, and a record player example for high school physics.

Simple Harmonic Motion from Circular 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。

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SC.912.P.10

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Simple Harmonic Motion from Circular Motion

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