Accélération angulaire

L'accélération angulaire est la vitesse à laquelle augmente la vitesse de rotation.

Données clés
Unités SI	rad⋅s⁻²
Dimension	T⁻²
Nature	Grandeur vectorielle intensive
Symbole usuel	α, ${\dot {\omega }}$ , ${\ddot {\theta }}$ ,
Lien à d'autres grandeurs	${\vec {\alpha }}$ = ${\partial \over \partial t}$ ${\vec {\omega }}$
Conjuguée	Moment d'inertie

En physique, l'accélération angulaire est la variation de la vitesse angulaire au cours du temps. En unités dérivées du Système international, l'accélération angulaire s'exprime en radians par seconde carrée (rad/s²).

L'accélération angulaire est une grandeur physique fondamentale pour caractériser le mouvement de rotation.

Énoncé

L'accélération est la première dérivée par rapport au temps (dérivée temporelle) de la vitesse angulaire, et la seconde dérivée temporelle de la position angulaire.

Si ${\vec {\omega }}$ est la vitesse angulaire et ${\vec {\theta }}$ la position angulaire, l'accélération angulaire ${\vec {\alpha }}$ est :

{\vec {\alpha }}={\frac {\mathrm {d} {\vec {\omega }}}{\mathrm {d} t}}={\frac {\mathrm {d} ^{2}{\vec {\theta }}}{\mathrm {d} t^{2}}}

L'accélération angulaire est pour le mouvement de rotation l'homologue de l'accélération linéaire pour le mouvement de translation.

Article détaillé : Analogie entre rotation et translation.

Accélérations tangentielle et centripète

L'accélération angulaire d'un corps est liée à ses accélérations tangentielle et centripète. Pour déterminer l'accélération tangentielle d'un corps, il suffit de multiplier son accélération angulaire par la mesure du rayon du cercle qui forme sa trajectoire^[1].

a_{\mathrm {t} }=r\alpha

Principe fondamental de la dynamique

L'accélération angulaire est l'une des variables de la deuxième loi de Newton appliquée en dynamique de rotation.

Ainsi, on peut déterminer le total des moments de forces ( ${\vec {\tau }}$ ) qui sont appliqués sur un corps à l'aide de l'accélération angulaire ( ${\vec {\alpha }}$ ) de celui-ci et de son moment d'inertie (I). La sommation de tous les moments de force est équivalente au produit du moment d'inertie du corps par son accélération angulaire lorsque le corps est rigide et que la rotation s'effectue autour d'un axe de rotation fixe^[2].

\sum {\vec {\tau }}=\mathrm {I} {\vec {\alpha }}

À-coup angulaire

L'à-coup angulaire ${\vec {\jmath }}_{\mathrm {a} }$ est la dérivée par rapport au temps de l'accélération angulaire :

{\vec {\jmath }}_{\mathrm {a} }={\frac {\mathrm {d} }{\mathrm {d} t}}{\vec {\alpha }}

Exemples

La vitesse angulaire est considérée positive lorsque le corps tourne en sens trigonométrique par rapport au référentiel du-dit solide.

Roue en rotation autour d'un axe fixe

Posons une roue tournant dans le sens positif d'un référentiel donné. On dira que la vitesse et l'accélération angulaire sont parallèles lorsque la vitesse augmente puisqu'elles seront toutes deux positives. À l'inverse, elles seront anti-parallèles si la vitesse diminue car celle-ci restera positive alors que l'accélération angulaire deviendra négative^[3].

Notes et références

↑ Benson 2009, p. 326
↑ Benson 2009, p. 337 - 340 (Expression valable lorsque l'axe de rotation est l'un des axes principaux d'inertie du corps considéré.)
↑ Kane et Sternheim 1986, p. 95

Annexes

Bibliographie

: document utilisé comme source pour la rédaction de cet article.

Harris Benson (trad. Marc Séguin, Benoît Villeneuve, Bernard Marcheterre et Richard Gagnon), Physique 1 Mécanique, Édition du Renouveau Pédagogique, 2009, 4^e éd., 465 p.
Joseph W. Kane et Morton M. Sternheim (adaptation Michel Delmelle), Physique, Interédition, 1986, 775 p.

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