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

じょうろがフックからぶら下がっていると考えてください。ピボットポイントからずれると、単純な振り子のように振動します。じょうろは物理的な振り子の一例です。
これは、全重量が重心に作用し、ピボットポイントを中心に振動するようにモデル化できます。ピボットポイントと重心の間の距離をLとします。
振動は重力によって生じる復元トルクによるもので、これは計算できます。振動角が小さい場合、トルクは近似されます。
また、振り子の慣性モーメントと角加速度の観点から記述することもできます。
2つの式は、質量が慣性モーメントに置き換えられ、力定数が3つの項の積に置き換えられた単純な調和運動の方程式を与えます。
したがって、角周波数を決定し、期間を導き出すことができます。
単純な振り子の場合、慣性モーメントは、その期間でおなじみの表現を与えます。

<ol>
	<li>ヤング、HD、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:セクション14.6;455-456ページ。</li>
	<li>OpenStaxです。(2019). 大学物理学 Vol.1.【Web版】。<a href="https://openstax.org/books/university-physics-volume-1/pages/1-introduction">https://openstax.org/books/university-physics-volume-1/pages/1-introduction から取得</a>:セクション15.4;766-768ページ。</li>
</ol>

物理的な振り子

剛体が固定されたピボット点から自由にぶら下がって変位すると、単純な振り子と同様に振動し、物理的な振り子として知られます。 物理的な振り子の周期と角周波数は、小角近似を使用し、ばね質量系で平行線を引くことによって取得されます。 小角近似 (sinθ=θ) は約 14° まで有効です。

複雑な系を扱う場合、慣性質量モーメントはピボット ポイントの周りの質量分布を表すため、重要なパラメータとなります。 慣性モーメントは、回転運動に対する物体の抵抗の尺度です。 複雑な質量分布を持つ振り子の場合、慣性モーメントの計算は困難で時間のかかる作業になる可能性があります。 ただし、重心フレームワークを使用すると、この計算を大幅に簡素化できます。 物体の複雑な質量分布とその結果として生じる質量慣性モーメントは、ピボット点の周りの慣性モーメントとピボット点と質量中心の間の距離という 2 つの項に単純化されます。 この単純化は、重心フレームワークを使用することの驚くべき威力を示しています。

実際の物理的な振り子の場合は、単振り子の慣性モーメントの式を使用することで、理想的な単振り子に帰着することを示すことができます。

物理的な振り子には便利な用途があります。 極端な状況では、強風や地震活動により、超高層ビルは最大 20 Hz の周波数で最大 2 メートル揺れる可能性があります。 いくつかの企業が、超高層ビルの屋上に設置する物理的な振り子を開発しました。 超高層ビルが右に揺れると、振り子は左に揺れて揺れが小さくなります。

物理的な振り子は、重力加速度の測定にも使用できます。

剛体が固定されたピボットポイントから自由にぶら下がっていて、変位すると、単純な振り子のように振動し、物理...

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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

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

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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Physical Pendulum Motion and Inertia

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