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

考虑月球距离地心为A，以恒定角速度作圆周运动。
设地球中心为位移-时间坐标系的原点。当月球运动至位置P时，其在x轴上的投影点P'与原点形成夹角Ф。
当月球在任意时刻 t 绕地球运动时，它所形成的角度为 ωt+Ф。根据月球在 x 轴或 y轴 上的投影，该投影的位置可用余弦函数或正弦函数表示。
月球的周期可以通过地球轨道的周长除以其速度来表示。根据能量守恒定律回顾速度方程并对其进行修正，即可确定月球投影的周期。
月球的速度方向沿切线方向，而月球的加速度方向则沿径向向内。
月球速度和加速度的 x 分量等于月球投影的速度和加速度。其大小可通过回顾速度和加速度方程得到。
如所观察到的，月球投影的周期、位置、速度和加速度方程与简谐振子的相应方程相似。
因此，沿圆周运动所在圆的直径方向投影的匀速圆周运动代表简谐运动。

简谐振动和匀速圆周运动

虽然简谐振动和匀速圆周运动可能是两个独立的概念，但它们相互关联且相互联系着。 简谐振动是系统中的振荡运动，其合力可以用胡克定律描述，而匀速圆周运动是物体以恒定速度沿圆周路径运动。 有一种简单的方法可以通过匀速圆周运动来产生简谐振动。 例如，考虑一个连接到均匀旋转的垂直转盘上的球，其阴影投射在地板上。 这里，影子的位置，也称为投影，进行简谐振动。 胡克定律通常描述匀速圆周运动（⍵常数），而不是具有较大可见位移的系统。 因此，观察匀速圆周运动的投影往往比观察精确的大型简谐振子更容易。 另一个例子是进行匀速圆周运动的电唱机。 考虑在转盘外边缘的一个点上连接一根销钉杆，并在销钉的另一端连接一支笔。 当电唱机转动时，笔就会移动。 如果我们将一张长纸拖到笔下，它的运动就会被捕获为波浪。 由此可知，简谐振动是匀速圆周运动沿发生圆周运动的圆直径的投影。 本文改编自 <a href="https://openstax.org/books/college-physics/pages/16-6-uniform-circular-motion-and-simple-harmonic-motion">Openstax，大学物理，第 16.6 节：匀速圆周运动和  简谐振动</a> 和 <a href="https://openstax.org/books/university-physics-volume-1/pages/15-3-comparing-simple-harmonic-motion-and-circular-motion">Openstax，大学物理第 1 卷，第 15.3 节：简谐运动和圆周运动的对比。</a>

尽管简谐运动与匀速圆周运动可能是两个独立的概念，但二者之间存在关联并相互联系。简谐运动……

Education

教育

JoVE Core

Physics

物理学

Unit 1

第一单元

Oscillations

振荡

KEY TERMS AND CONCEPTS

关键术语与概念

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

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Florida Department of Education

High School

Physical Science

SC.912.P.10

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Simple Harmonic Motion from Circular Motion

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