Name: physical pendulum motion and inertia
Uploaded: 2023-05-16T08:18:41.000Z
Duration: 1 min 6 s
Channel: JoVE

K12 Standards

Florida Department of Education

High School

Physical Science

SC.912.P.10

SC.912.P.10.6

Physical Pendulum

갈고리에 매달려 있는 물뿌리개를 생각해 보십시오. 피벗점에서 이동하면 단순 진자와 유사하게 진동합니다. 물뿌리개는 물리적 진자의 예입니다.
전체 무게가 피벗점을 중심으로 진동하는 질량 중심에서 작용하는 것으로 모델링할 수 있습니다. 피벗점과 질량 중심 사이의 거리를 L로 합니다.
진동은 계산할 수 있는 중력에 의해 생성된 복원 토크 때문입니다. 진동 각도가 작으면 토크가 근사화됩니다.
또한 진자의 관성 모멘트와 각가속도의 관점에서 쓸 수도 있습니다.
두 표현식은 단순 조화 운동에 대한 방정식을 제공하며, 질량은 관성 모멘트로 대체되고 힘 상수는 세 항의 곱으로 대체됩니다.
따라서 각 주파수를 결정할 수 있고 기간을 도출할 수 있습니다.
단순 진자의 경우 관성 모멘트는 해당 기간에 대한 친숙한 표현을 제공합니다.

<ol>
	<li>Young, H.D 및 Freedman, RA (2012). 현대 물리학을 가진 대학 물리학. 샌프란시스코, 캘리포니아: 피어슨: 섹션 14.6; 455-456쪽.</li>
	<li>오픈스택스. (2019). 대학 물리학 Vol. 1. [웹 버전]. 원본 주소 <a href="https://openstax.org/books/university-physics-volume-1/pages/1-introduction">https://openstax.org/books/university-physics-volume-1/pages/1-introduction</a>: section 15.4; 766-768쪽.</li>
</ol>

물리적 진자

강체는 고정된 피벗점에 자유롭게 매달려 있고 변위될 때 단순 진자와 유사하게 진동하며 이를 물리적 진자라고 합니다. 물리적 진자의 주기와 각진동수는 소각 근사법을 사용하고 스프링-질량 시스템과 평행선을 그려서 얻습니다. 작은 각도 근사(sinθ=θ)는 약 14°까지 유효합니다.

복잡한 시스템을 다룰 때 질량 관성 모멘트는 피벗점 주위의 질량 분포를 설명하므로 중요한 매개변수입니다. 관성 모멘트는 회전 운동에 대한 물체의 저항을 측정한 것입니다. 질량 분포가 복잡한 진자의 경우 관성 모멘트를 계산하는 것은 어렵고 시간이 많이 걸리는 작업일 수 있습니다. 그러나 질량 중심 프레임워크를 사용하면 이 계산을 크게 단순화할 수 있습니다. 몸체의 복잡한 질량 분포와 결과적인 질량 관성 모멘트는 피벗점에 대한 관성 모멘트와 피벗점과 질량 중심 사이의 거리라는 두 가지 용어로 단순화됩니다. 이러한 단순화는 질량 중심 프레임워크 사용의 놀라운 힘을 보여줍니다.

실제 물리 진자의 경우, 단진자의 관성 모멘트에 대한 표현을 이용하여 이상화된 단진자로 환원되는 것을 보여줄 수 있다.

물리 진자는 유용한 용도로 사용됩니다. 극한 상황에서는 강풍이나 지진 활동으로 인해 고층 건물이 최대 20Hz의 주파수로 최대 2미터까지 흔들릴 수 있습니다. 여러 회사에서 고층 건물 꼭대기에 배치되는 물리적 진자를 개발했습니다. 고층 건물이 오른쪽으로 흔들리면 진자가 왼쪽으로 흔들리면서 흔들림이 줄어듭니다.

물리 진자를 사용하여 중력으로 인한 가속도를 측정할 수도 있습니다.

강체가 고정된 피봇 포인트에서 자유롭게 매달려 있다가 변위되면 단순 진자와 유사하게 진동하며 물리적 물체로 알려져 있습니다.

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Education

교육

JoVE Core

Physics

물리학

Unit 1

제1단원

Oscillations

진동

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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 pivot point. Moment of inertia measures resistance to rotational motion. For a body with a complicated mass distribution, finding this value can take time.

The center of mass framework makes the calculation easier. It reduces the mass distribution problem to two terms: the moment of inertia about the pivot point and the distance from the pivot point to the center of mass. This approach shows how useful the center of mass model can be. A real physical pendulum can also be reduced to the ideal simple pendulum by using the simple pendulum moment of inertia expression.

Physical pendulums have practical uses. Very tall skyscrapers can sway up to two meters with a frequency of up to 20 Hz during strong winds or seismic activity. Some companies place physical pendulums on top of buildings to reduce that sway. As the skyscraper moves to one side, the pendulum swings to the other. Physical pendulums can also be used to measure the acceleration due to gravity.

Physical Pendulum Motion and Inertia

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

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

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...

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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

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...

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 a physical pendulum swings, how moment of inertia affects its motion, and how center of mass ideas help explain real-world uses.

Physical Pendulum Motion and Inertia | JoVE K12

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Physical Pendulum Motion and Inertia

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