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

Weźmy pod uwagę konewkę wiszącą na haczyku. Po przemieszczeniu od punktu obrotu oscyluje podobnie jak zwykłe wahadło. Konewka jest przykładem wahadła fizycznego.
Można go modelować jako cały jego ciężar działający na środek masy, który oscyluje wokół punktu obrotu. Niech odległość między punktem obrotu a środkiem masy wynosi L.
Oscylacja jest spowodowana przywracającym momentem obrotowym wytwarzanym przez grawitację, który można obliczyć. Jeśli kąt oscylacji jest mały, moment obrotowy jest przybliżony.
Można go również zapisać w kategoriach momentu bezwładności wahadła i przyspieszenia kątowego.
Te dwa wyrażenia dają równanie prostego ruchu harmonicznego, w którym masa jest zastąpiona momentem bezwładności, a stała siły jest zastąpiona iloczynem trzech wyrazów.
W ten sposób można określić częstotliwość kątową i wyprowadzić okres czasu.
Dla prostego wahadła moment bezwładności daje znajome wyrażenie dla danego okresu.

<ol>
	<li>Young, H.D i Freedman, R.A. (2012). Fizyka uniwersytecka z fizyką współczesną. San Francisco, Kalifornia: Pearson: sekcja 14.6; strony 455-456.</li>
	<li>Protokół OpenStax. (2019). Fizyka uniwersytecka Vol. 1. [Wersja internetowa]. Pobrane z <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>: sekcja 15.4; strony 766-768.</li>
</ol>

wahadło fizyczne

Kiedy ciało sztywne zwisa swobodnie z ustalonego punktu obrotu i ulega przemieszczeniu, oscyluje podobnie do prostego wahadła i jest znane jako wahadło fizyczne. Okres i częstotliwość kątową wahadła fizycznego oblicza się, stosując przybliżenie małych kątów i rysując równoleżniki za pomocą układu masa-sprężyna. Przybliżenie małego kąta (sinθ=θ) obowiązuje do około 14'.

W przypadku skomplikowanych układów moment bezwładności masy jest ważnym parametrem, ponieważ opisuje rozkład masy wokół punktu obrotu. Moment bezwładności jest miarą oporu obiektu wobec ruchu obrotowego. W przypadku wahadła o skomplikowanym rozkładzie masy obliczenie momentu bezwładności może być zadaniem trudnym i czasochłonnym. Jednakże, korzystając ze środka masy, możemy znacznie uprościć te obliczenia. Skomplikowany rozkład masy ciała i wynikający z niego moment bezwładności masy zostaje uproszczony do dwóch pojęć: momentu bezwładności względem punktu obrotu oraz odległości punktu obrotu od środka masy. To uproszczenie pokazuje niezwykłą moc wykorzystania struktury środka masy.

Można wykazać, że przypadek prawdziwego wahadła fizycznego sprowadza się do wyidealizowanego wahadła prostego, używając wyrażenia opisującego moment bezwładności wahadła prostego.

Wahadła fizyczne mają przydatne zastosowania. W ekstremalnych warunkach drapacze chmur mogą kołysać się do dwóch metrów z częstotliwością do 20 Hz z powodu silnego wiatru lub aktywności sejsmicznej. Kilka firm opracowało wahadła fizyczne umieszczane na szczytach drapaczy chmur. Gdy drapacz chmur kołysze się w prawo, wahadło odchyla się w lewo, zmniejszając kołysanie.

Wahadła fizyczne można również wykorzystać do pomiaru przyspieszenia grawitacyjnego.

Wahadło fizyczne

Kiedy sztywne ciało zwisa swobodnie ze stałego punktu obrotu i jest przemieszczone, oscyluje podobnie do prostego wahadła i jest znane jako fizyczne ...

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Education

JoVE Core

Physics

Fizyka

Unit 1

Jednostka 1

Oscillations

Oscylacje

KEY TERMS AND CONCEPTS

Kluczowe terminy i pojęcia

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

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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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