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2 Basic Concepts From Physics (Motion about a fixed axis…
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
\(\mathbf{a}=\dot{\mathbf{v}}=\frac{d}{dt}(\dot{s}\mathbf{T})=\frac{d\dot{s}}{dt}\mathbf{T}+\dot{s}\frac{d\mathbf{T}}{dt}=\)
\(=\ddot{s}\mathbf{T}+\dot{s}\frac{ds}{dt}\frac{d\mathbf{T}}{ds}=\)
\(=\ddot{s}\mathbf{T}+\kappa\dot{s}^{2}\mathbf{N}\)
\(\mathbf{v}=\dot{s}\mathbf{T}\)
\[\kappa=\frac{\mathbf{v}\cdot\mathbf{a}^{\bot}}{\left|\mathbf{v}\right|^{3}}=\frac{\dot{x}\ddot{y}-\dot{y}\ddot{x}}{(\dot{x}^{2}+\dot{y}^{2})^{3/2}}\]
Polar
\[\mathbf{r}=r\mathbf{R}\]
\[\mathbf{v}=\dot{r}\mathbf{R}+r\dot{\theta}\mathbf{P}\]
\[\mathbf{a}=(\ddot{r}-r\dot{\theta}^{2})\mathbf{R}+(r\ddot{\theta}+2\dot{r}\dot{\theta})\mathbf{P}\]
motion without external forces
i 2D
: N(t) = T(t) roterad 90 grader counter-clockwise
Spatial motion
Cylindrical
\[\mathbf{r}=r\mathbf{R}+z\mathbf{k}\]
\[\mathbf{v}=\dot{r}\mathbf{R}+r\dot{\theta}\mathbf{P}+\dot{z}\mathbf{k}\]
\[\mathbf{a}=(\ddot{r}-r\dot{\theta}^{2})\mathbf{R}+(r\ddot{\theta}+2\dot{r}\dot{\theta})\mathbf{P}+\ddot{z}\mathbf{k}\]
(samma som 2d polar förutom z-komponenten)
Cartesian
\[\kappa=\frac{\left|\mathbf{v}\times\mathbf{a}\right|}{\left|\mathbf{v}\right|^{3}}\]
\[\mathbf{v}\times\mathbf{a}=v\frac{d\left|\mathbf{v}\right|}{dt}\mathbf{T}\times\mathbf{T}+\left|v\right|^{3}\kappa\mathbf{T}\times\mathbf{N}=\left|v\right|^{3}\kappa\mathbf{B}\]
\[\mathbf{a}=\frac{d\mathbf{v}}{dt}=\frac{d\left|v\right|}{dt}\mathbf{T}+\left|\mathbf{v}\right|\frac{d}{dt}\mathbf{T}=\frac{\mathbf{v}\cdot\mathbf{a}}{\left|\mathbf{v}\right|}\mathbf{T}+\left|v\right|^{2}\kappa\mathbf{N}\]
\[\mathbf{v}=\frac{d\mathbf{r}}{dt}=\frac{d\mathbf{r}}{ds}\,\frac{ds}{dt}=v\mathbf{T}\]
...
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
\(\mathbf{R}=(\cos\theta)\boldsymbol{\xi}+(\sin\theta)\boldsymbol{\eta}\)
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 \(\boldsymbol{\alpha}(t)=\dot{\mathbf{\sigma}}(t)\mathbf{D}\)
\(\mathbf{r}(t)=r_{0}\mathbf{R}(t)+h_{0}\mathbf{D}\)
\(\mathbf{v}(t)=\mathbf{w}\times\mathbf{r}\)
\(\mathbf{a}(t)=-r_{0}\sigma^{2}\mathbf{R}+\boldsymbol{\alpha}\times\mathbf{r}\)
Motion about a moving axis
\(R(t)=I+(\sin(\theta(t)))\textrm{Skew}(\mathbf{D}(t))+(1-cos(\theta(t)))\textrm{Skew}(\mathbf{D}(t))^{2}\)
\(\mathbf{w}=\dot{\theta}\mathbf{D}+(\sin\theta)\dot{\mathbf{D}}+(\cos\theta-1)\dot{\mathbf{D}}\times\mathbf{D}\)