2 Basic Concepts From Physics

Rigid Body

Classification
"By the region that its mass lives in"

Single particle

of mass m

at the single location x

Continuous material

infinitely many particles

in bounded region R

Kinematics

Planar motion

Cartesian

$\mathbf{a}=\dot{\mathbf{v}}=\frac{d}{dt}(\dot{s}\mathbf{T})=\frac{d\dot{s}}{dt}\mathbf{T}+\dot{s}\frac{d\mathbf{T}}{dt}=$

$=\ddot{s}\mathbf{T}+\dot{s}\frac{ds}{dt}\frac{d\mathbf{T}}{ds}=$

$=\ddot{s}\mathbf{T}+\kappa\dot{s}^{2}\mathbf{N}$

$\mathbf{v}=\dot{s}\mathbf{T}$

motion without external forces

i 2D: N(t) = T(t) roterad 90 grader counter-clockwise

\[\kappa=\frac{\mathbf{v}\cdot\mathbf{a}^{\bot}}{\left|\mathbf{v}\right|^{3}}=\frac{\dot{x}\ddot{y}-\dot{y}\ddot{x}}{(\dot{x}^{2}+\dot{y}^{2})^{3/2}}\]

Polar

\[\mathbf{r}=r\mathbf{R}\]

\[\mathbf{v}=\dot{r}\mathbf{R}+r\dot{\theta}\mathbf{P}\]

\[\mathbf{a}=(\ddot{r}-r\dot{\theta}^{2})\mathbf{R}+(r\ddot{\theta}+2\dot{r}\dot{\theta})\mathbf{P}\]

Spatial motion

Cylindrical

Cartesian

\[\kappa=\frac{\left|\mathbf{v}\times\mathbf{a}\right|}{\left|\mathbf{v}\right|^{3}}\]

\[\mathbf{v}\times\mathbf{a}=v\frac{d\left|\mathbf{v}\right|}{dt}\mathbf{T}\times\mathbf{T}+\left|v\right|^{3}\kappa\mathbf{T}\times\mathbf{N}=\left|v\right|^{3}\kappa\mathbf{B}\]

\[\mathbf{a}=\frac{d\mathbf{v}}{dt}=\frac{d\left|v\right|}{dt}\mathbf{T}+\left|\mathbf{v}\right|\frac{d}{dt}\mathbf{T}=\frac{\mathbf{v}\cdot\mathbf{a}}{\left|\mathbf{v}\right|}\mathbf{T}+\left|v\right|^{2}\kappa\mathbf{N}\]

\[\mathbf{v}=\frac{d\mathbf{r}}{dt}=\frac{d\mathbf{r}}{ds}\,\frac{ds}{dt}=v\mathbf{T}\]

...

\[\mathbf{r}=r\mathbf{R}+z\mathbf{k}\]

\[\mathbf{v}=\dot{r}\mathbf{R}+r\dot{\theta}\mathbf{P}+\dot{z}\mathbf{k}\]

\[\mathbf{a}=(\ddot{r}-r\dot{\theta}^{2})\mathbf{R}+(r\ddot{\theta}+2\dot{r}\dot{\theta})\mathbf{P}+\ddot{z}\mathbf{k}\]

(samma som 2d polar förutom z-komponenten)

Spherical

$\mathbf{a}=\left((\rho\ddot{\theta}+2\dot{\rho}\dot{\theta})\sin\phi+2\rho\dot{\theta}\dot{\phi}\cos\phi\right)\mathbf{P}$

\[\mathbf{v}=(\rho\dot{\theta}\sin\phi)\mathbf{P}+(-\rho\dot{\phi})\mathbf{Q}+(\dot{\rho})\mathbf{R}\]

Q är vinkelrät mot båda: $(-\cos\theta\cos\phi,-\sin\theta\cos\phi,\sin\phi)$

P ligger i planet och är vinkelrät till R: $(-\sin\theta,\cos\theta,0)$

R är riktningsvektorn mot r: $(\cos\theta\sin\phi,\sin\theta\sin\phi,\cos\phi)$

$+\left(\rho(\dot{\theta}^{2}\sin\phi\cos\phi-\ddot{\phi})-2\dot{\rho}\dot{\phi}\right)\mathbf{Q}$

$+\left(\ddot{\rho}-\rho(\dot{\phi}^{2}+\dot{\theta}^{2}\sin^{2}\phi)\right)\mathbf{R}$

Moving frame: {${\mathbf{r}(t);\,\mathbf{P}(t),\,\mathbf{Q}(t),\,\mathbf{R}(t)}$}

Motion about a fixed axis

$\mathbf{R}=(\cos\theta)\boldsymbol{\xi}+(\sin\theta)\boldsymbol{\eta}$

Motion about a moving axis

angular speed $\sigma(t)=\dot{\theta}(t)$

Välj D som uppåtvektor, och ha R i $\boldsymbol{\xi}\,\boldsymbol{\eta}$-planet

angular velocity $\mathbf{w}(t)=\sigma(t)\mathbf{D}$

angular acceleration $\boldsymbol{\alpha}(t)=\dot{\mathbf{\sigma}}(t)\mathbf{D}$

$\mathbf{r}(t)=r_{0}\mathbf{R}(t)+h_{0}\mathbf{D}$

$\mathbf{v}(t)=\mathbf{w}\times\mathbf{r}$

$\mathbf{a}(t)=-r_{0}\sigma^{2}\mathbf{R}+\boldsymbol{\alpha}\times\mathbf{r}$

$R(t)=I+(\sin(\theta(t)))\textrm{Skew}(\mathbf{D}(t))+(1-cos(\theta(t)))\textrm{Skew}(\mathbf{D}(t))^{2}$

$\mathbf{w}=\dot{\theta}\mathbf{D}+(\sin\theta)\dot{\mathbf{D}}+(\cos\theta-1)\dot{\mathbf{D}}\times\mathbf{D}$