« Formulaire de mécanique » : différence entre les versions

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{{FormulesPhysique}}
 
== [[Cinématique]] : le rayon vecteur et ses dérivées successives ==
=== En [[coordonnées cartésiennes]] ===
<math>\overrightarrow{OM}=x\overrightarrow{u_x}+y\overrightarrow{u_y}+z\overrightarrow{u_z}</math>
 
:<math>\boldsymbol r =x \boldsymbol u_x + y \boldsymbol u_y + z \boldsymbol u_z</math>
<math>\overrightarrow{v}(M)=\frac{ \text{d} \overrightarrow{OM} }{ \text{d} t }=\frac{\text{d} x}{\text{d} t}\overrightarrow{u_x}+\frac{\text{d} y}{\text{d} t}\overrightarrow{u_y}+\frac{\text{d} z}{\text{d} t}\overrightarrow{u_z}</math>
La vitesse du point situé en '''''r''''' s'écrit
 
:<math>\overrightarrow{a}(M)=\frac{\text{d}boldsymbol \overrightarrow{v} (M)}{\text{d}boldsymbol r) t}=\frac{ \text{d}^2 \overrightarrow{OM}boldsymbol r}{ \text{d} t^2 }=\frac{\text{d}^2 x}{\text{d} t^2} \overrightarrow{boldsymbol u_x} + \frac{\text{d}^2 y}{\text{d} t^2} \overrightarrow{boldsymbol u_y} + \frac{\text{d}^2 z}{\text{d} t^2} \overrightarrow{boldsymbol u_z}</math>,
et l'accélération
:<math>\boldsymbol a(\boldsymbol r)=\frac{\text{d} \boldsymbol v}{\text{d} t}=\frac{\text{d}^2 \boldsymbol r}{\text{d} t^2}=\frac{\text{d}^2 x}{\text{d} t^2} \boldsymbol u_x + \frac{\text{d}^2 y}{\text{d} t^2} \boldsymbol u_y + \frac{\text{d}^2 z}{\text{d} t^2} \boldsymbol u_z</math>.
 
=== En [[coordonnées cylindriques]] ===
<math>\overrightarrow{OM}=\rho\overrightarrow{u_\rho}+z\overrightarrow{u_z}</math>
 
:<math>\boldsymbol r=\rho \boldsymbol u_\rho+z \boldsymbol u_z</math>
<math>\overrightarrow{v}(M)=\frac{\text{d} \overrightarrow{OM}}{\text{d} t}= \frac{\text{d} \rho}{\text{d} t}\overrightarrow{u_\rho}+\rho\frac{\text{d} \phi}{\text{d} t}\overrightarrow{u_{\phi}}+\frac{\text{d} z}{\text{d}t}\overrightarrow{u_z}</math>
:<math>\overrightarrow{boldsymbol v}(M)=\frac{ \text{d} \overrightarrow{OM}boldsymbol r}{ \text{d} t }= \frac{\text{d} x\rho}{\text{d} t} \overrightarrow{u_x}boldsymbol u_\rho+\rho\frac{\text{d} y\varphi}{\text{d} t} \overrightarrow{u_y}boldsymbol u_\varphi +\frac{\text{d} z}{\text{d} t} \overrightarrow{boldsymbol u_z}</math>.
:<math> \boldsymbol \overrightarrow{a}(M) =\frac{\text{d} \overrightarrow{boldsymbol v}(M)}{\text{d} t}=\frac{\text{d}^2 \overrightarrow{OM}boldsymbol r}{\text{d}t^2}=\left(\frac{\text{d}^2 \rho}{\text{d} t^2}-\rho\left(\frac{\text{d} \phivarphi}{\text{d} t}\right)^2\right) \overrightarrow{boldsymbol u_\rho}+\left(2\frac{\text{d} \rho}{\text{d} t}\frac{\text{d} \phivarphi}{\text{d} t}+\rho\frac{\text{d}^2 \phi}{\text{d}t^2}\right) \overrightarrow{boldsymbol u_{\phi}}varphi+\frac{\text{d}^2 z}{\text{d}t^2} \overrightarrow{boldsymbol u_z}</math>.
 
Ces formules sont basées sur le fait que la dérivée temporelle de deux des vecteurs de base est non nulle :
<math> \overrightarrow{a}(M)=\frac{\text{d} \overrightarrow{v}(M)}{\text{d} t}=\frac{\text{d}^2 \overrightarrow{OM}}{\text{d}t^2}=\left(\frac{\text{d}^2 \rho}{\text{d} t^2}-\rho\left(\frac{\text{d} \phi}{\text{d} t}\right)^2\right)\overrightarrow{u_\rho}+\left(2\frac{\text{d} \rho}{\text{d} t}\frac{\text{d} \phi}{\text{d} t}+\rho\frac{\text{d}^2 \phi}{\text{d}t^2}\right)\overrightarrow{u_{\phi}}+\frac{\text{d}^2 z}{\text{d}t^2}\overrightarrow{u_z}</math>
:<math> \frac{\text{d} \boldsymbol u_\rho}{\text{d} t}=\frac{\text{d}\varphi}{\text{d}t} \boldsymbol u_\varphi</math>,
 
En utilisant: <math>\displaystyle \frac{\text{d} \overrightarrow{boldsymbol u_\rho}varphi}{\text{d} t}=-\frac{\text{d}\phivarphi}{\text{d}t}\overrightarrow{u_{\phi}}</math> et <math> \frac{\text{d}boldsymbol \overrightarrow{u_{\phi}}}{\text{d} t}=-\frac{\text{d}\phi}{\text{d}t}\overrightarrow{u_\rho}</math>.
 
=== En [[coordonnées sphériques]] ===
 
:<math>\overrightarrow{OM}boldsymbol r=r \overrightarrow{boldsymbol u_r}</math>,
:<math>\overrightarrow{boldsymbol v}(M) =\frac{\text{d} \overrightarrow{OM}boldsymbol r}{\text{d} t}= \frac{\text{d} \rhor}{\text{d} t} \overrightarrow{u_\rho}boldsymbol u_r+\rhor\frac{\text{d} \phitheta}{\text{d} t} \overrightarrow{boldsymbol u_{\phi}}theta+r \frac{\text{d} z\varphi}{\text{d}t}\overrightarrow{u_z}sin \theta \boldsymbol u_\varphi</math>;
 
:<math>\overrightarrow{v}(M)boldsymbol a =\frac{\text{d} \overrightarrow{OM}boldsymbol v }{\text{d} t}=\frac{\text{d}^2 \boldsymbol r}{\text{d}t^2}=a_r \overrightarrow{boldsymbol u_r}+ra_\frac{\text{d}theta \theta}{\text{d}boldsymbol t}\overrightarrow{u_{\theta}}+r \frac{\text{d}a_\varphi}{ \text{d}t}\sinboldsymbol \theta\overrightarrow{u_\varphi}</math>,
 
<math>\overrightarrow{a}(M)=\frac{\text{d} \overrightarrow{v}(M)}{\text{d} t}=\frac{\text{d}^2 \overrightarrow{OM}}{\text{d}t^2}=a_r\overrightarrow{u_r}+a_\theta\overrightarrow{u_\theta}+a_\varphi\overrightarrow{u_\varphi}</math>
 
avec:
:<math>a_\thetaa_r=\left( r \frac{\text{d}^2 \theta2r}{\text{d}t^2} +2\frac{\text{d}-r}{\text{d}t} left(\frac{\text{d} \theta}{\text{d} t}-\right)^2+r\left( \frac{\text{d}\varphi}{\text{d}t} \right)^2\sin \theta \cos ^2\theta\right)</math>,
 
:<math>a_ra_\theta=\left( r \frac{\text{d}^2r2 \theta}{\text{d}t^2}- +2\frac{\text{d}r}{\left(text{d}t} \frac{\text{d} \theta}{\text{d} t}\right)^2+-r\left( \frac{\text{d}\varphi}{\text{d}t} \right)^2\sin^2 \theta \cos \theta\right)</math>
:<math>a_\varphi=\left( r \frac{\text{d}^2 \varphi}{\text{d}t^2}\sin \theta +2\frac{\text{d}r}{\text{d}t} \frac{\text{d} \varphi}{\text{d} t}\sin \theta + 2r\frac{\text{d} \varphi}{\text{d} t}\frac{\text{d} \theta}{\text{d} t}\cos \theta\right)</math>.
 
<math>a_\theta=\left( r \frac{\text{d}^2 \theta}{\text{d}t^2} +2\frac{\text{d}r}{\text{d}t} \frac{\text{d} \theta}{\text{d} t}-r\left( \frac{\text{d}\varphi}{\text{d}t} \right)^2\sin \theta \cos \theta\right)</math>
 
<math>a_\varphi=\left( r \frac{\text{d}^2 \varphi}{\text{d}t^2}\sin \theta +2\frac{\text{d}r}{\text{d}t} \frac{\text{d} \varphi}{\text{d} t}\sin \theta + 2r\frac{\text{d} \varphi}{\text{d} t}\frac{\text{d} \theta}{\text{d} t}\cos \theta\right)</math>
 
== Changement de référentiel ==
2

modifications