Name: stable equilibrium through potential energy
Uploaded: 2023-09-22T11:49:37.000Z
Duration: 1 min 16 s
Channel: JoVE

K12 Standards

Next Generation Science Standards

High School

Physical Science

Energy

HS-PS3-5

potential-energy-criterion-for-equilibrium

Potential-Energy Criterion for Equilibrium

Konumu tek bir bağımsız değişken tarafından tanımlanan sürtünmesiz bir yay-kütle sistemi düşünün. Burada potansiyel enerji, yerçekimi ve elastik potansiyel enerjilerinin toplamıdır ve değişiminin negatifi sistem üzerinde yapılan işe eşittir.
Sistem dengedeyse ve sanal bir yer değiştirmeye maruz kalırsa, yapılan iş sanal çalışma ile değiştirilebilir.
Sanal çalışma ilkesine göre, tüm sanal yer değiştirmeler için iş sıfırdır ve bu nedenle potansiyel enerjideki değişim de sıfırdır. Küçük sanal yer değiştirmeler için, potansiyel enerji, konum koordinatı ile ilk türevi cinsinden ifade edilir.
Sanal yer değiştirme sıfır olmadığından, potansiyel enerjinin ilk türevi sıfır olmalıdır. Dolayısıyla, bir sistem potansiyel enerjisinin ilk türevi sıfır olduğunda dengededir.
Bu kriteri yay-kütle sistemine uygulamak, denge konumunu verir.
Potansiyel enerji birkaç bağımsız değişkene bağlıysa, her bir koordinat için potansiyel enerjinin kısmi türevi denge için sıfır olmalıdır.

<ol>
	<li>Hibbeler, R.C. (2016). Mühendislik Mekaniği: Statik. On Dördüncü Baskı, New Jersey: Pearson. Sayfa 600-602.</li>
	<li>Meriam, J.L., Kraige, L.G. ve Bolten, J.N.(2016). Mühendislik Mekaniği: Statik ve Dinamik. Sekizinci Baskı, Singapur: John Wiley & Sons. Sayfa 415-416.</li>
</ol>

denge için potansiyel enerji kriteri

Potansiyel enerji veya potansiyel fonksiyon, mekanik bir sistemin dengesini belirlemede önemli bir rol oynar. Bir sistem hem yerçekimi kuvvetine hem de elastik kuvvetlere maruz kaldığında, sistemdeki potansiyel fonksiyon yerçekimi ve elastik potansiyel enerjisinin cebirsel toplamı olarak ifade edilebilir. Bir sistem denge durumunda ise ve küçük bir miktarla yerinden kaydırılırsa, sistem üzerinde yapılan iş, sistemdeki potansiyel enerjinin başlangıcından son durumuna değişiminin negatifidir. Eğer sistem gerçek yerine sanal bir yer değiştirme geçirirse, o zaman sanal iş ilişkisi sistemin potansiyel enerjisindeki değişimin tüm sanal yer değişmeleri için sıfır olması gerektiğini önerir. Bu, mekanik bir sistemin denge konfigürasyonunun, sistemdeki toplam potansiyel enerjinin bir sabit değeri olduğu anlamına gelir. Konfigürasyonu tanımlayan tek bir değişken olan x için potansiyel enerji ve türevleri sürekli bir fonksiyon olduğunda, denge durumu şu şekilde yazılabilir:

<img alt="Equation 1" src="/files/ftp_upload/14359/PT_14359_Equation_1.png" style="text-align: center; height: 30px;" />

Yukarıdaki denklem, bir mekanik sistemin toplam potansiyel enerjisinin türevidir ve sıfır olduğunda sistem dengededir. Birden fazla dereceye sahip sistemler için, her koordinata göre potansiyel enerjinin parsiyel türevidi dengede sıfır olmalıdır.

Denge konfigürasyonunda kararlılık, potansiyel enerjinin o konfigürasyonda minimum olması durumunda devam eder. Minimum potansiyel enerjiyi sağlamak için potansiyel fonksiyonun yer değiştirme koordinatlarına göre ikinci türevidi pozitif olmalıdır. Bu, bir sistemin kararlı denge için ikinci temel koşuldur.

Denge için Potansiyel Enerji Kriteri

Potansiyel enerji veya potansiyel fonksiyon, mekanik bir sistemin kararlılığını belirlemede önemli bir rol oynar. Bir sistem her ikisine de tabi tutulursa ...

Education

JoVE Core

Mechanical Engineering

Makine Mühendisliği

Unit 1

Ünite 1

Virtual Work

Sanal İş

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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 an imagined small shift rather than an actual motion, the virtual work relation shows that the change in potential energy must be zero for every virtual displacement.

This means the equilibrium configuration is a state where the total potential energy has a stationary value. For a system with one degree of freedom, where the configuration is described by a single variable x and the potential energy is a continuous function of x, equilibrium occurs when the derivative of total potential energy with respect to x is zero. For systems with several degrees of freedom, the partial derivative of potential energy with respect to each coordinate must also be zero.

Stability depends on whether the potential energy is at a minimum at the equilibrium position. To make sure the equilibrium is stable, the second derivative of the potential function with respect to the displacement coordinates must be positive. This positive second-derivative condition is the second requirement for stable equilibrium.

Stable Equilibrium Through Potential Energy

stable-equilibrium-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 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-configuration-problem-solving

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...

key terms and concepts

Learn how gravitational and elastic potential energy determine equilibrium, why the first derivative is zero, and how a positive second derivative shows stability.

Stable Equilibrium 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.

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Stable Equilibrium Through Potential Energy

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