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

K12 Standards

north-carolina-k-12-science-standard-course-of-study

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

지구 중심에서 A 거리에 있는 달이 일정한 각속도로 원을 그리며 회전한다고 가정해 보십시오.
지구의 중심을 변위-시간 좌표계의 원점으로 합니다. 달이 P 위치로 이동하면 x축의 투영 P'가 각도 Ф를 만듭니다.
달이 t 시간 동안 지구 주위를 더 멀리 이동함에 따라 각도 ωt+Ф를 만듭니다. x축 또는 y축에 대한 달의 투영을 기반으로 투영의 위치는 코사인 또는 사인 함수로 표시할 수 있습니다.
달의 주기는 달의 속도에 대한 지구 궤도의 둘레에 의해 주어질 수 있습니다. 에너지 보존에서 속도 방정식을 불러오고 수정하면 달의 투영 주기가 결정됩니다.
달의 속도는 접선으로 작용하는 반면 달의 가속도는 방사상으로 안쪽을 향합니다.
달의 속도와 가속도의 x 성분은 달의 투영 속도와 가속도와 같습니다. 그들의 크기는 속도와 가속도 방정식을 불러와서 얻습니다.
관찰된 바와 같이, 달의 투영의 시간 주기, 위치, 속도 및 가속도 방정식은 단순한 고조파 발진기의 방정식과 유사합니다.
그러므로, 원운동이 발생하는 원의 지름을 따라 균일한 원운동의 투영은 단순 조화 운동을 나타냅니다.

단순 조화 운동과 균일 원운동

단순조화운동과 등속원운동은 서로 다른 두 가지 개념일 수 있지만 서로 연관되어 연결되어 있습니다. 단순 조화 운동은 훅의 법칙으로 알짜 힘을 설명할 수 있는 시스템의 진동 운동인 반면, 등속 원 운동은 일정한 속도로 원형 경로를 따라 움직이는 물체의 운동입니다.

등속 원운동을 사용하여 단순 조화 운동을 생성하는 쉬운 방법이 있습니다. 예를 들어, 균일하게 회전하는 수직 턴테이블에 부착된 공의 그림자가 바닥에 투영되는 경우를 생각해 보세요. 여기서 투영이라고도 하는 그림자의 위치는 단조파 운동을 수행합니다. Hooke의 법칙은 일반적으로 눈에 보이는 큰 변위가 있는 시스템보다는 균일한 원 운동(⍵ 상수)을 설명합니다. 따라서 균일한 원운동의 투영을 관찰하는 것이 정밀한 대규모 단순 조화 발진기를 관찰하는 것보다 더 쉬운 경우가 많습니다.

또 다른 예는 균일한 원운동을 하는 레코드 플레이어입니다. 턴테이블 바깥쪽 가장자리의 한 지점에 부착된 다웰 막대와 다웰의 다른 쪽 끝에 부착된 펜을 생각해 보십시오. 레코드 플레이어가 회전하면 펜이 움직입니다. 펜 아래에 긴 종이를 드래그하면 그 움직임이 파동으로 포착됩니다. 이로부터 단조화운동은 원운동이 일어나는 원의 지름을 따라 등속 원운동을 투영하는 것임을 이해할 수 있다.

이 텍스트는 <a href="https://openstax.org/books/college-physics/pages/16-6-uniform-circular-motion-and-simple-에서 수정되었습니다. harmonic-motion">Openstax, 대학 물리학, 섹션 16.6: 균일 원형 운동 및  단순 조화 운동</a> 및 <a href="https://openstax.org/books/university-physics-volume-1/pages/15-3-comparing- simple-harmonic-motion-and-circular-motion">Openstax, 대학 물리학 1권, 섹션 15.3: 단순 조화 운동과 원형 운동 비교</a>

단순 조화 운동과 균일 원 운동은 두 개의 별개의 개념일 수 있지만 서로 상관 관계가 있고 상호 연결됩니다. 단순 고조파 ...

Education

교육

JoVE Core

Physics

물리학

Unit 1

제1단원

Oscillations

진동

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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...

/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

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

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Simple Harmonic Motion from Circular Motion

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