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

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HS-PS3-5

potential-energy-criterion-for-equilibrium

Potential-Energy Criterion for Equilibrium

Si consideri un sistema molla-massa senza attrito la cui posizione è definita da una variabile indipendente. Qui, l'energia potenziale è la somma delle energie potenziali gravitazionali ed elastiche, e il negativo della sua variazione è uguale al lavoro svolto sul sistema.
Se il sistema è in equilibrio e subisce uno spostamento virtuale, il lavoro svolto può essere sostituito dal lavoro virtuale.
Dal principio del lavoro virtuale, il lavoro è zero per tutti gli spostamenti virtuali, e quindi anche la variazione dell'energia potenziale è zero. Per piccoli spostamenti virtuali, l'energia potenziale è espressa in termini della sua derivata prima con la coordinata di posizione.
Poiché lo spostamento virtuale non è zero, la derivata prima dell'energia potenziale deve essere zero. Quindi, un sistema è in equilibrio quando la derivata prima della sua energia potenziale è zero.
Applicando questo criterio al sistema molla-massa si ottiene la sua posizione di equilibrio.
Se l'energia potenziale dipende da più variabili indipendenti, allora la derivata parziale dell'energia potenziale per ciascuna coordinata deve essere zero per l'equilibrio.

<ol>
	<li>Hibbeler, R.C. (2016). Ingegneria Meccanica: Statica. Quattordicesima edizione, New Jersey: Pearson. Pagine 600-602.</li>
	<li>Meriam, J.L., Kraige, L.G. e Bolten, J.N. (2016). Meccanica ingegneristica: statica e dinamica. Ottava edizione, Singapore: John Wiley & Sons. Pagine 415-416.</li>
</ol>

criterio della energia potenziale per l'equilibrio

L'energia potenziale o la funzione potenziale, svolge un ruolo essenziale nel determinare la stabilità di un sistema meccanico. Se un sistema è soggetto sia a forze gravitazionali che elastiche, la funzione potenziale del sistema può essere espressa come la somma algebrica dell'energia potenziale gravitazionale ed elastica. Se il sistema è in equilibrio e viene spostato di una piccola quantità, il lavoro compiuto sul sistema è uguale al negativo della variazione dell'energia potenziale del sistema, dalla posizione iniziale a quella finale. Se il sistema subisce uno spostamento virtuale invece di uno spostamento reale, la relazione del lavoro virtuale suggerisce che la variazione dell'energia potenziale del sistema dovrebbe essere zero per tutti gli spostamenti virtuali. Questo significa che la configurazione di equilibrio di un sistema meccanico è quella in cui l'energia potenziale totale del sistema ha un valore stazionario. Per un sistema con un grado di libertà in cui l'energia potenziale e le sue derivate sono una funzione continua di una singola variabile x, che descrive la configurazione, la condizione di equilibrio può essere scritta come:

<img alt="Equation 1" src="/files/ftp_upload/14359/PT_14359_Equation_1.png" style="text-align: center; height: 30px;" />

L'equazione sopra afferma che un sistema meccanico è in equilibrio quando la derivata della sua energia potenziale totale è zero. Per sistemi con più gradi di libertà, la derivata parziale dell'energia potenziale rispetto a ciascuna coordinata, deve essere zero per l'equilibrio.

La stabilità nella configurazione di equilibrio, rimane finché la sua energia potenziale è minima in quella configurazione. Per garantire una minima energia potenziale, la seconda derivata della funzione potenziale rispetto alle coordinate di spostamento, deve essere positiva. Questa è la seconda condizione essenziale per l'equilibrio stabile di un sistema.

Criterio della energia potenziale per l'equilibrio

L'energia potenziale o funzione potenziale gioca un ruolo essenziale nel determinare la stabilità di un sistema meccanico. Se un sistema è sottoposto a entrambi...

Education

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JoVE Core

Mechanical Engineering

Ingegneria Meccanica

Unit 1

Unità 1

Virtual Work

Lavoro virtuale

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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

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Stable Equilibrium Through Potential Energy

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