Mechanics topics include Mechanics of Particles, Mechanics of Rigid Bodies, Mechanics of Continous Media and Special Relativity.
Mechanics of Rigid Bodies
Mechanics of Continous Media
Special Relativity
Preliminary Mathematics
Plane-Polar Coordinate System
finding unit vectors $\hat{r}$ and $\hat{\theta}$
$x = r \cos \theta$
$y = r \sin \theta$
$\tan \theta = \frac{y}{x}$
finding unit vectors $\hat{r}$ and $\hat{\theta}$
$\vec{r} = r \cos \theta \hat{i} + r \sin \theta \hat{j}$
$\hat{r} = \frac{\vec{r}}{|\vec{r}|} = \cos \theta \hat{i} + \sin \theta \hat{j}$
and
$\vec{\theta} = \frac{d\vec{r}}{d\theta} = - r \sin \theta \hat{i} + r \cos \theta \hat{j}$
$\hat{\theta} = \frac{\vec{\theta}}{|\vec{\theta}|} = - \sin \theta \hat{i} + \cos \theta \hat{j}$
$\frac{d\hat{r}}{dt} = \frac{d(\cos \theta \hat{i} + \sin \theta \hat{j})}{dt}$
$\Rightarrow \dot{\hat{r}} = - \sin \theta \cdot \dot{\theta} \hat{i} + \cos \theta \cdot \dot{\theta} \hat{j}$
$ \dot{\hat{r}} = \dot{\theta} (- \sin \theta \hat{i} + \cos \theta \hat{j})$
$\boxed{\dot{\hat{r}} = \dot{\theta} \cdot \hat{\theta}}$
similarly
$\boxed{\dot{\hat{\theta}} = - \dot{\theta} \cdot \hat{r}}$
finding velocity and acceleration from $\vec{r}$
we may use $\vec{r} = r \cdot \hat{r}$
velocity $\vec{v} = \frac{d\vec{r}}{dt} = \frac{d(r \cdot \hat{r})}{dt}$
$\Rightarrow \vec{v}=\frac{d r}{d t} \cdot \hat{r}+\frac{r d \hat{r}}{d t} = \dot{r} \hat{r}+r(\dot{\theta} \cdot \hat{\theta}) $
$\Rightarrow \boxed{ \vec{v}=\dot{r} \hat{r}+r \dot{\theta} \hat{\theta} }$
similarly acceleretion, $\vec{a}=\frac{d \vec{v}}{d t}$
$ \Rightarrow \boxed{ \vec{a}=\left(\ddot{r}-r \dot{\theta}^2\right) \hat{r}+(r \ddot{\theta}+2 \dot{r} \dot{\theta}) \hat{\theta} }$
unit vectors $\hat{r}$ and $\hat{\theta}$ are perpendicular in its direction.
Spherical polar coordinates
Three coordinates $r, \theta, \phi$ are used to describe the position of any point.
$r \rightarrow$ radius vector
$\theta \rightarrow$ co-latitude
$\phi \rightarrow$ azimuthal angle
the plane polar coordinates can be written using spherical polar coordinates
$x = r \sin \theta \cos \phi$
$y = r \sin \theta \sin \phi$
$z = r \cos \theta$
$\vec{r} = r \sin \theta \cos \phi \hat{i} + r \sin \theta \sin \phi \hat{j} + r \cos \theta \hat{k}$
using,
$\vec{\theta} = \frac{d \vec{r}}{d \theta}$
and
$\vec{\phi} = \frac{d \vec{r}}{d \phi}$
we get,
$\hat{r} = \frac{\vec{r}}{|\vec{r}|}$
$\hat{\theta} = \frac{\vec{\theta}}{|\vec{\theta}|} = \frac{(d\vec{r}/d\theta)}{|\vec{\theta}|}$
$\hat{\phi} = \frac{\vec{\phi}}{|\vec{\phi}|} = \frac{(d\vec{r}/d\phi)}{|\vec{\phi}|}$
thus, the spherical coordinates unit vectors are given by,
$\hat{r} = \sin \theta \cos \phi \hat{i} + \sin \theta \sin \phi \hat{j} + \cos \theta \hat{k}$
$\hat{\theta} = \cos \theta \cos \phi \hat{i} + \cos \theta \sin \phi \hat{j} - \sin \theta \hat{k}$
$\hat{\phi} = - \sin \phi \hat{i} + \cos \phi \hat{j}$
and the spherical polar coordinates can be written using plane polar coordinates,
$|\vec{r}| = \sqrt{x^2 + y^2 + z^2}$
$|\vec{\theta}| = \tan ^{-1} (\frac{\sqrt{x^2 + y^2}}{z})$
$|\vec{\phi}| = \tan ^{-1} (\frac{y}{x})$
The unit vectors $\hat{r}, \hat{\theta}, \hat{\phi}$ are orthogonal.
$\boxed{ \hat{r} \times \hat{\theta} = \hat{\phi} \hat{\theta} \times \hat{\phi} = \hat{r} \hat{\phi} \times \hat{r} = \hat{\theta} }$
Line element - general infinitesimal displacement $d \vec{l}$ will be given by
$d \vec{l} = dr \hat{r} + r d \theta \space \hat{\theta} + r \sin \theta \space d \phi \space \hat{\phi}$
Volume element - infinitesimal volume element $d \vec{l}$ is the product of $dl_{r}, dl_{\theta}, dl_{\phi}$, given by
$dV = r^2 \sin \theta d \space \theta dr \space d \phi$