Name: finding force from potential energy
Uploaded: 2023-04-30T10:57:22.000Z
Duration: 1 min 13 s
Channel: JoVE

K12 Standards

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North Carolina K-12 Science Standard Course of Study

9-12

Physics

Energy

PS.Phy.6.2

force-and-potential-energy-in-one-dimension

Force and Potential Energy in One Dimension

Le forze conservative possono essere espresse in termini di energia potenziale.
Consideriamo un corpo che si muove lungo una linea retta a causa di una forza conservativa che è una funzione della posizione.
Ora, il lavoro svolto dalla forza conservatrice è uguale al negativo della variazione della sua energia potenziale.
Il lavoro svolto è anche uguale al prodotto della forza e dello spostamento causato dalla forza.
Pertanto, la forza sul corpo è uguale al negativo della variazione dell'energia potenziale divisa per lo spostamento. 
Il segno negativo indica che la forza conservativa trascina il sistema a un'energia potenziale inferiore.
Per spostamenti molto piccoli, è possibile stabilire l'espressione per forza ed energia potenziale in una dimensione.
Per la verifica della relazione tra forza ed energia potenziale, si consideri l'esempio dell'energia potenziale elastica. Quando viene sostituita nella relazione tra forza ed energia, la formula per la legge di Hooke è stabilita.

forza ed energia potenziale in una dimensione

La forza può essere calcolata dall'espressione dell'energia potenziale, che è una funzione di posizione. La componente di una forza conservativa, in una certa direzione, è uguale al negativo della derivata dell'energia potenziale corrispondente rispetto allo spostamento in quella direzione. Per le zone dove l'energia potenziale cambia rapidamente con lo spostamento, il lavoro svolto dalla forza è massimo. Inoltre, quando si applica una forza lungo l'asse delle coordinate positive, l'energia potenziale diminuisce con l'aumentare dello spostamento. Dunque, la forza e l'energia potenziale derivata hanno segno opposto rispetto allo spostamento.

Per esempio, se l'energia potenziale di una particella che subisce un moto unidimensionale lungo l'asse x è:

<img alt="Equation1" src="/files/ftp_upload/12677/12677_Equation_1.png" style="height: 32px;" />

Dove c = 8 N/m3, quale sarà la forza per l'energia potenziale data? Qui, la grandezza conosciuta è l'energia potenziale. Quella sconosciuta e che dev'essere calcolata è la quantità di foza.

<img alt="Equation1" src="/files/ftp_upload/12677/12677_Equation_2.png" style="height: 32px;" />

Aggiungendo il valore dell'energia potenziale nell'espressione precedente, la forza lungo la direzione dell'asse x viene calcolata essere -cx3. Sostituendo il valore di c, la forza è calcolata come -8x3.

La funzione dell'energia potenziale per la forza tra due atomi in una molecola diatomica è approssimativamente data da:

<img alt="Equation1" src="/files/ftp_upload/12677/12677_Equation_3.png" style="height: 32px;" />

Dove a e b sono costanti e x è la distanza tra gli atomi. Se l'energia di dissociazione della molecola è:

<img alt="Equation1" src="/files/ftp_upload/12677/12677_Equation_4.png" style="height: 14px;" />

Qual è il valore di D? Qui, la grandezza conosciuta è l'energia potenziale. La grandezza sconosciuta è l'energia di dissociazione della molecola. Sostituendo la funzione del potenziale d'azione, otteniamo l'espressione per la forza. Visto che la forza netta all'equilibrio è zero, otteniamo:

<img alt="Equation1" src="/files/ftp_upload/12677/12677_Equation_5.png" style="height: 32px;" />

Ora, il valore di x6 viene sostituito nell'espressione per l'energia potenziale. Visto che all'infinito l'energia potenziale è zero, l'energia di dissociazione di una molecola è calcolata per essere b2/4a.

Questo testo è adattato da <a href="https://openstax.org/books/university-physics-volume-1/pages/8-4-potential-energy-diagrams-and-stability">Openstax, University Physics Volume 1, Section 8.4: Potential Energy Diagrams and Stability.</a>

Forza ed energia potenziale in una dimensione

La forza può essere calcolata dall'espressione dell'energia potenziale, che è una funzione della posizione. La componente di una forza conservatrice, in un particolare ...

Education

Didattica

JoVE Core

Fisica

Unit 1

Unità 1

Potential Energy and Energy Conservation

Energia Potenziale e Conservazione dell'Energia

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

finding force from potential energy

Force can be found from potential energy when the energy is written as a function of position. For a conservative force, the force component in a given direction equals the negative derivative of the matching potential energy with respect to displacement. This means force and the slope of potential energy have opposite signs.

Where potential energy changes quickly with displacement, the force and the work done are largest. When a force acts along the positive coordinate axis, potential energy decreases as displacement increases. In one-dimensional motion, this relationship is often used to move from a known potential energy function to the force it produces.

For example, if a particle moves along the x-axis and its potential energy is given with c = 8 N/m^3, the force can be calculated from that expression. The potential energy is the known quantity, and the force is the unknown quantity. Differentiating the energy function gives the x-direction force as -cx^3, which becomes -8x^3 after substituting the value of c.

The same idea also applies to the force between two atoms in a diatomic molecule. Their potential energy can be written approximately as a function of distance x, with a and b as constants. If the dissociation energy is needed, the force is first found from the potential energy, and the equilibrium point is used because the net force is zero there. Then the value of x^6 is substituted into the energy expression, and since the potential energy at infinity is zero, the dissociation energy is b^2/4a.

Finding Force from Potential Energy

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Gravitational Energy from Height and Weight

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force and potential energy in one dimension

Force can be found from potential energy when the energy is written as a function of position. For a conservative force, the force component in a given direction equals the negative derivative of the matching potential energy with respect to displacement. This means force and the slope of potential energy have opposite signs. Where potential energy changes quickly with displacement, the force and the work done are largest. When a force acts along the positive coordinate axis, potential energy...

force and potential energy in three dimensions

force-and-potential-energy-in-three-dimensions

Gradient Force from 3D Potential Energy

Gradient force from 3D potential energy shows how a conservative force changes along the x, y, and z axes. The potential energy function U depends on all three spatial coordinates. Each force component is linked to how U changes with position along one coordinate. In one dimension, force can be written as the negative ratio of the change in potential energy to the displacement along that axis. For very small displacements, these ratios become derivatives. When a function has several variables,...

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energy diagrams - ii

Stability in Potential Energy Diagrams

Potential energy diagrams show how stable a system is at different positions. Their shape, or topology, helps identify equilibrium points. An equilibrium point is a place where the system can stay in balance because the net force is zero. At a local minimum, the potential energy is lowest. The system tends to remain there unless a net force acts on it. The slope at the minimum is zero. The slope is positive on both sides, so the force acts as a restoring force and pushes the system back. A...

key terms and concepts

Learn how to calculate force from potential energy in one dimension, including the sign rule, a particle example, and dissociation energy for atoms.

Finding Force from Potential Energy | JoVE K12

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Finding Force from Potential Energy

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