Name: dimensional checks for physics equations
Uploaded: 2023-05-16T08:18:40.000Z
Duration: 1 min 8 s
Channel: JoVE

Supposons qu’un pendule simple de masse m soit attaché à une corde de longueur L, oscillant sous l’influence de la gravité g. Quelle est la forme de l'équation pour la période de temps du pendule ?
Dans un premier temps, identifiez et énumérez les variables impliquées dans le problème. La période de temps T peut être exprimée comme le produit de ces variables, chacune élevée à un exposant inconnu. Ici, k est une constante sans dimension.
En excluant la constante sans dimension, on obtient une équation reliant les dimensions des variables à la période.
Maintenant, en égalisant les exposants des dimensions des deux côtés et en résolvant les équations, les valeurs des exposants inconnus sont déterminées.
En substituant les exposants, on obtient l’expression finale de la période de temps, qui est un produit de la constante k et de la racine carrée de la longueur sur l’accélération gravitationnelle.
L’une des limites de l’analyse dimensionnelle est qu’elle ne nous permet pas de déterminer la valeur de la constante sans dimension k.

<ol>
	<li>OpenStax. (2019). Physique universitaire Vol. 1. [Version Web]. Récupéré de <a href="https://openstax.org/books/university-physics-volume-1/pages/1-4-dimensional-analysis">https://openstax.org/books/university-physics-volume-1/pages/1-4-dimensional-analysis</a> ; section 1.4 ; Pages 22 à 25.</li>
</ol>

résolution de problèmes : analyse dimensionnelle

Toute équation mathématique reliant des quantités physiques séparées et distinctes doit être dimensionnellement cohérente, ce qui implique qu'elle doit respecter deux règles. Pour cette raison, le concept de dimension est crucial. La première règle stipule que les expressions d'une équation de part et d'autre d'une égalité doivent avoir exactement la même dimension, c'est-à-dire que des quantités de même dimension peuvent être ajoutées ou supprimées. La deuxième règle stipule que toutes les fonctions mathématiques populaires, telles que les fonctions exponentielles, logarithmiques et trigonométriques, doivent avoir des arguments sans dimension dans une équation.

Une équation qui enfreint l'une de ces deux règles est dimensionnellement incohérente, et ne peut donc pas représenter avec précision une loi physique. L'analyse dimensionnelle peut aider à mémoriser les différentes lois de la physique, à vérifier les erreurs ou les fautes de frappe algébriques, et même à spéculer sur la forme que pourraient prendre les lois de la physique futures.

Les quantités de base peuvent être utilisées pour créer toutes les quantités physiques souhaitées. Une quantité est exprimée comme le produit de différentes puissances des quantités de base lorsqu'elle est exprimée en termes des quantités de base. La dimension de la quantité dans cette base est l'exposant d'une quantité de base qui apparaît dans l'équation.

Considérons la quantité physique de force, qui est définie comme la masse multipliée par l'accélération. L'accélération est calculée comme le changement de vitesse divisé par un intervalle de temps, tandis que la longueur divisée par l'intervalle de temps donne la vitesse. En conséquence, la force a les dimensions suivantes : une en masse, une en longueur et moins deux en temps.

Résolution de problèmes : Analyse dimensionnelle

Toute équation mathématique qui relie des quantités physiques distinctes séparées doit être dimensionnellement cohérente, ce qui implique qu’elle doit respecter deux ...

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Education

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Physics

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Unit 1

Unité 1

Units, Dimensions, and Measurements

Unités, dimensions et mesures

KEY TERMS AND CONCEPTS

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Problem Solving: Dimensional Analysis

dimensional checks for physics equations

Dimensional analysis helps check whether physics equations make sense. It compares the dimensions on each side of an equation and shows whether the units are consistent. This method is useful for spotting algebra errors, typos, and for remembering physical laws.

A valid equation must follow two rules. First, both sides of an equality must have the same dimension, so only quantities with matching dimensions can be added or subtracted. Second, common math functions such as exponential, logarithmic, and trigonometric functions must use dimensionless arguments.

If an equation breaks either rule, it is dimensionally inconsistent. That means the equation cannot correctly represent a physical law. Dimensional analysis also helps scientists think about the form future laws of physics might take.

Base quantities are the building blocks for other physical quantities. A quantity can be written as a product of powers of the base quantities. In that form, the dimension of a quantity is given by the exponent of each base quantity.

Force is a useful example. Force is defined as mass multiplied by acceleration. Acceleration is the change in velocity divided by time, and velocity is length divided by time. From these relationships, force has dimensions of mass to the first power, length to the first power, and time to the minus second power.

Dimensional Checks for Physics Equations

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key terms and concepts

Learn how dimensional analysis checks physics equations, keeps units consistent, and reveals the dimensions of force from mass, length, and time.

Dimensional Checks for Physics Equations | JoVE K12

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Dimensional Checks for Physics Equations

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