Name: equilibrium stability through potential energy
Uploaded: 2023-09-22T11:49:38.000Z
Duration: 1 min 13 s
Channel: JoVE

K12 Standards

Texas Essential Knowledge and Skills

High School

Physics

PHYS.1

IPC.1.G

stability-of-equilibrium-configuration-problem-solving

Stability of Equilibrium Configuration: Problem Solving

Jantına bağlı bir bilye ile yaylı kasnak sistemine bağlı bir disk düşünün.
Disk döndürülür ve sadece iş yapan kuvvetlerin serbest cisim diyagramı çizilir. Dönmelerdeki sapmalar koordinat eksenine göre gösterilir.
Sistemin denge konumlarını ve kararlılıklarını belirlemek için, elastik ve yerçekimi potansiyel enerji fonksiyonlarının toplamı elde edilir.
Yayın ve bilyenin dikey sapması belirlenir ve potansiyel enerji fonksiyonuna ikame edilir.
Dengede, potansiyel enerji fonksiyonunun ilk türevi sıfırdır. Sayısal değerlerin çözülmesi ve yerine konması, dönme açısı cinsinden denge konumlarını verir.
Denge açılarının kararlılığı, potansiyel enerjinin ikinci türevinden belirlenir.
İkinci türev negatifse, denge kararsızdır. Bu açıda, hafif bir hareket sistemi dönmeye ayarlar.
Pozitifse, denge kararlıdır, bu da dönme altında sistemin bu açıda durduğu anlamına gelir.

<ol>
	<li>Hibbeler, R.C. (2016). Mühendislik Mekaniği: Statik. On Dördüncü Baskı, New Jersey: Pearson. Bölüm 11.7, sayfa 603-606.</li>
	<li>Bira, F.P., Johnston, E.R., Mazurek, D.F., Cornwell, P.J. ve Öz, B.P. (2016). Mühendisler için Vektör Mekaniği. On Birinci Baskı, New York: McGraw-Hill Eğitimi. Sayfa 601.</li>
</ol>

denge konfigürasyonunun kararlılığı: problemlerin Çözülmesi

Denge konfigürasyonunun kararlılığı, fizik, mühendislik ve diğer ilgili alanlarda önemli bir kavramdır. Basit bir ifadeyle, bir nesnenin veya sistemin, bozulduktan sonra denge pozisyonuna geri dönme eğilimini ifade eder. Denge konfigürasyonunun kararlılığı, sistemin potansiyel enerji fonksiyonunu göz önünde bulundurarak ve denge noktasına yakın davranışını inceleyerek analiz edilebilir.

Denge konfigürasyonunun kararlılığı bağlamında problem çözme birkaç adımdan oluşur. İlk olarak, sistemin denge noktaları belirlenmelidir. Bunun için potansiyel enerji fonksiyonunun türevidini sıfıra eşitlemek ve denklemi sağlayan bağımsız değişkenlerin değerlerini bulmak gerekmektedir.

Denge noktaları belirlendikten sonra, bir sonraki adım onların kararlılığını incelemektir. Bunun için potansiyel enerji fonksiyonunun ikinci türevi her denge noktasında değerlendirilir. Eğer ikinci türev pozitif ise, denge kararlıdır ve sistem bozulduğunda denge pozisyonuna geri döner. Tersine, ikinci türev negatif ise, denge kararsızdır ve sistem bozulduğunda denge pozisyonundan uzaklaşır.

Önemli bir nokta, bir sistemin birden fazla denge noktası olabileceğidir ve her birinin kararlılığının ayrı ayrı analiz edilmesi gerekmektedir. Ayrıca, denge konfigürasyonunun kararlılığı, sönümleme veya dış etkiler gibi çeşitli faktörlerden etkilenebilir.

Denge Konfigürasyonunun Kararlılığı: Problemlerin Çözülmesi

Denge konfigürasyonlarının kararlılığı fizik, mühendislik ve diğer ilgili alanlarda önemli bir kavramdır. Basit bir ifadeyle, şu anlama gelir...

Education

JoVE Core

Mechanical Engineering

Makine Mühendisliği

Unit 1

Ünite 1

Virtual Work

Sanal İş

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

equilibrium stability through potential energy

Stability of equilibrium in physics and engineering can be studied by using a system’s potential energy. An equilibrium configuration is a state where the system can remain at rest. Stability describes whether the system tends to return to that state after a small disturbance.

Problem solving begins by finding the equilibrium points of the system. This is done by setting the derivative of the potential energy function equal to zero and solving for the independent variable values that satisfy the equation. A system may have more than one equilibrium point, so each point must be checked on its own.

The next step is to test stability at each equilibrium point. For this, the second derivative of the potential energy function is evaluated at the equilibrium point. If the second derivative is positive, the equilibrium is stable. If it is negative, the equilibrium is unstable and the system moves away after being disturbed.

Other factors can also affect stability. Damping and external forces may change how a system behaves near equilibrium. Because of this, equilibrium analysis is an important problem-solving tool in physics, engineering, and related fields.

Equilibrium Stability Through Potential Energy

equilibrium-stability-through-potential-energy

work

Work in Force and Motion

Work connects force, motion, and energy in mechanics. It happens when a force acts on an object and the object moves along the line of action of that force. Work can also be described as energy transfer caused by applying a force, which leads to a change in the object’s motion or position. Work is calculated as the force multiplied by the part of the displacement in the same direction as the force. It can also be written as the dot product of the applied force and the object’s displacement. In...

work of a couple moment

Couple Moment Work in Rotation

Mechanical engineering studies motion, energy, and force, and it uses the idea of a couple moment to describe how two equal and opposite forces act on a rigid body. A couple moment is created when those forces act at two points on the body that are separated by a distance. When a rigid body moves because of a couple, its displacement can be split into two parts. First, the two points move by the same amount to their final positions. Then the body rotates about one of those points. During the...

virtual work

Virtual Work in Force Equilibrium

Virtual work is a mechanics principle used to check equilibrium with imaginary motion. If a body is in static or dynamic equilibrium, the total virtual work from external forces and couple moments must be zero for any virtual displacement. A virtual displacement is an imagined small movement. It can be a shift or a rotation. The virtual work done by a force equals the force dotted with the virtual displacement in the force’s direction. For a couple moment, virtual rotational work is found by...

virtual work for a system of connected rigid bodies

virtual-work-for-a-system-of-connected-rigid-bodies

Finding Forces with Virtual Work in Linked Bodies

Virtual work helps find forces in a system of connected rigid bodies. It is a useful method when several links move together and the system is in equilibrium. In that case, the total virtual work is zero. To analyze the system, start with a free-body diagram. Use one independent coordinate to describe the link configuration. Then mark the deflected position that comes from a positive virtual displacement. This makes the small change in position clear before any calculations begin. Next,...

principle of virtual work: problem solving

principle-of-virtual-work-problem-solving

Virtual Work for Finding Unknown Forces

The principle of virtual work helps find unknown forces and moments in equilibrium problems. It is an important idea in mechanics and engineering. The method uses a small, hypothetical motion called a virtual displacement. If the system is in equilibrium, the work done by the forces during that displacement is zero. To apply this principle, the positions of the internal and external forces must be written in terms of one common coordinate parameter. That parameter can be an angle, a distance,...

mechanical efficiency of real machines

Friction’s Role in Machine Efficiency

Mechanical efficiency shows how well a machine turns input work into output work. It is calculated as the ratio of output work to input work. An ideal machine has an efficiency of one, which means the input work equals the output work. Real machines are never perfectly ideal because some energy is lost. Friction is one of the main causes of that loss. In a toggle vise used to compress wooden blocks, the sliding block rubs against the horizontal plane and creates friction. To find the...

conservative forces

Conservative Forces and Path-Independent Work

Conservative forces are forces whose work depends only on an object’s starting and ending positions. The path between those positions does not change the amount of work done. This idea is important in mechanical engineering because it helps describe how energy moves through a system. A conservative force also conserves energy in the sense that its work is path independent. Gravity is one example. Electrostatic forces and springs modeled by Hooke’s law are also examples. A ball rolling down a...

potential energy

Gravitational and Elastic Energy Stored

Gravitational and elastic potential energy are forms of stored energy from conservative forces. A conservative force, such as gravity or an elastic force, gives a body the capacity to do work. This capacity is measured as potential energy and depends on the body’s position relative to a fixed reference point, or datum. For gravitational potential energy, the datum is the level where the energy is taken as zero. If a body is above this reference plane, its weight has positive gravitational...

potential-energy criterion for equilibrium

potential-energy-criterion-for-equilibrium

Stable Equilibrium Through Potential Energy

Potential energy helps determine when a mechanical system is stable at equilibrium. When a system is acted on by gravitational and elastic forces, its total potential function is the algebraic sum of gravitational potential energy and elastic potential energy. If the system is already in equilibrium and is given a small displacement, the work done on the system equals the negative of the change in potential energy between the starting and ending positions. For a virtual displacement, which is...

stability of equilibrium configuration

Types of Equilibrium in Mechanical Systems

Types of equilibrium in mechanical systems are stable, neutral, and unstable. These three cases help explain how a structure or object responds when it is pushed slightly away from balance. They are important for predicting whether a system will return to its original position, stay where it is, or move farther away. Stable equilibrium occurs when a system returns toward its original position after a small displacement. In this case, the potential energy is at a minimum. A cantilever beam...

one-degree-of-freedom system

Spring-Mass Motion and Energy Stability

A one-degree-of-freedom system uses a single independent variable to describe its state and behavior. In mechanical engineering, this kind of model helps explain how parts in a larger system respond to force. It also gives engineers a way to predict motion before a design is built in real life. A simple harmonic oscillator is one example of a one-degree-of-freedom system. A mass attached to a spring moves along one axis, and that position determines the system’s state. The spring force makes...

stability of equilibrium configuration: problem solving

Stability of equilibrium in physics and engineering can be studied by using a system’s potential energy. An equilibrium configuration is a state where the system can remain at rest. Stability describes whether the system tends to return to that state after a small disturbance. Problem solving begins by finding the equilibrium points of the system. This is done by setting the derivative of the potential energy function equal to zero and solving for the independent variable values that satisfy...

key terms and concepts

Learn how to find equilibrium points and test stability with potential energy, first and second derivatives, and signs of stable or unstable motion.

Equilibrium Stability Through Potential Energy | JoVE K12

JoVE Core Makine Mühendisliği, denge, kuvvetler, kafes kiriş analizi ve yapısal mekanik gibi temel konuları kapsayan kısa, anlaşılması kolay animasyonlu videolar sunar. Gerçek dünyadaki uygulamalarını keşfedin ve bilim insanlarının önderlik ettiği deneylerle bu kavramların uygulanmasına tanık olun.

mechanical engineering

Chemistry and Physics

IPC.1

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PHYS.1.A

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Experimental Designs

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Observation and Background Research
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from-theory-to-design-the-role-of-creativity-in-designing-experiments

From Theory to Design: The Role of Creativity in Designing Experiments

Source: Laboratories of Gary Lewandowski, Dave Strohmetz, and Natalie Ciarocco&#8212;Monmouth University

Research studies come into being when a ...

Apply scientific practices to plan and conduct descriptive, comparative, and experimental investigations, and use engineering practices to design solutions to problems

PHYS.1.B

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Guidelines in Case of a Laboratory Emergency

Robert M. Rioux &amp; Zhifeng Chen, Pennsylvania State University, University Park, PA
The most common laboratory emergencies include chemical spills, ...

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Source: Ali Bazzi, Department of Electrical Engineering, University of Connecticut, Storrs, CT.
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Robert M. Rioux &amp; Suprita Jharimune, Pennsylvania State University, University Park, PA
Among the many hazards present in the laboratory, electrical ...

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Robert M. Rioux, Pennsylvania State University, University Park, PA
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Use appropriate safety equipment and practices during laboratory, classroom, and field investigations as outlined in Texas Education Agency-approved safety standards

PHYS.1.C

Common Lab Glassware and Uses

Source: Laboratory of Dr. Neal Abrams &#8212; SUNY College of Environmental Science and Forestry
Glassware is a regular appearance in the professional ...

Lab Techniques

Source: Smaa Koraym at Johns Hopkins University, MD, USA

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	Density is a measure of how compact a substance or ...

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	Error can be introduced in any experiment, even ...

Cleaning Glassware

Source: Vy M. Dong and Daniel Kim, Department of Chemistry, University of California, Irvine, CA
Organic synthesis is about transforming a readily ...

Proper Lab Notebook Keeping

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	Nitration of Methyl BenzoateExpand
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sample-preparation-for-analytical-characterization

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Source: Laboratory of Dr. B. Jill Venton - University of Virginia
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Use appropriate tools such as balances, ballistic carts or equivalent, batteries, computers, constant velocity cars, convex lenses, copper wire, discharge tubes with power supply (H, He, Ne, Ar), data acquisition probes and software, dynamics and force demonstration equipment, electrostatic generators, electrostatic kits, friction blocks, graph paper, graphing technology, hand-held visual spectroscopes, inclined planes, iron filings, lab masses, laser pointers, magnets, magnetic compasses, metric rulers, motion detectors, multimeters (current, voltage, resistance), optics bench, optics kit, photogates, plane mirrors, prisms, protractors, pulleys, resistors, rope or string, scientific calculators, stopwatches, springs, spring scales, switches, tuning forks, wave generators, or other equipment and materials that will produce the same results.

PHYS.1.D

Units and Standards of Measurement

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International System of Units

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How Data are Classified: Numerical Data

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Quantitative Analysis

Quantitative analysis is a technique for measuring the amount of specific constituents in a sample. When the sample&#39;s composition is unknown, ...

Qualitative Analysis

Qualitative analysis is the process of identifying elements, ions, or compounds in an unknown sample. It is the first and most fundamental type of ...

Collect quantitative data using the International System of Units (SI) and qualitative data as evidence

PHYS.1.E

Review and Preview

Data are individual items of information obtained from a population or sample. Data may be classified as qualitative (categorical), quantitative ...

Ogive Graph

An ogive graph is sometimes called a cumulative frequency polygon. It is one type of frequency polygon that shows cumulative frequency. In other words, ...

Histogram

The histogram is a graphical representation in the x-y form of data distribution in a data set. The horizontal x-axis is labeled with what the data ...

Relative Frequency Histogram

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Scatter Plot

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Bar Graph

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Multiple Bar Graph

As the name suggests, a multiple bar graph is the same as a bar graph but has multiple bars to depict relationships between different data values. One can ...

Pareto Chart

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Pie Chart

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Life Tables

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PHYS.1.F

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Scientific and engineering practices. The student, for at least 40% of instructional time, asks questions, identifies problems, and plans and safely conducts classroom, laboratory, and field investigations to answer questions, explain phenomena, or design solutions using appropriate tools and models. The student is expected to:

Equilibrium Stability Through Potential Energy

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