PX285 - A2b - angular momentum

eg: $\vec{ω} = ω \hat{n}$ , $\vec{r}$ at an angle $θ$

\begin{aligned} \vec{v} & = \vec{ω} \times \vec{r} \\ = \vec{ω} \times (r \cos θ \hat{n} + r \sin θ \hat{x}) \\ = \vec{ω} \times \hat{x} r \sin θ \\ = \hat{v} r \sin θ ω \end{aligned}

$\vec{ω} \times \vec{x} = ω \hat{v}$

using the fact that all mass elements, $m_{i}$ (atoms), in a solid body rotate about the same axis if the body is rotated:

{\vec{v}}_{i} = \vec{ω} \times {\vec{r}}_{i}

\begin{aligned} \vec{L} & = \sum_{i} m_{i} {\vec{r}}_{i} \times (\vec{ω} \times {\vec{r}}_{i}) \\ = \sum_{i} m_{i} [\vec{ω} ({\vec{r}}_{i} \cdot {\vec{r}}_{i}) - {\vec{r}}_{i} ({\vec{r}}_{i} \cdot \vec{ω})] \end{aligned}

\begin{aligned} ⟹ L_{x} & = \sum_{i} m_{i} [ω_{x} ({\vec{r}}_{i} \cdot {\vec{r}}_{i}) - x_{i} (x_{i} ω_{x} + y_{i} ω_{y} + z_{i} ω_{z})] \\ = ω_{x} \sum_{i} m_{i} ({\vec{r}}_{i} \cdot {\vec{r}}_{i} - x_{i}^{2}) + ω_{y} \sum_{i} m_{i} (- x_{i} y_{i}) + ω_{z} \sum_{i} m_{i} (- x_{i} z_{i}) \end{aligned}

let:

\begin{aligned} I_{x x} & = \sum_{i} m_{i} ({\vec{r}}_{i} \cdot {\vec{r}}_{i}), \\ I_{x y} & = \sum_{i} m_{i} (- x_{i} y_{i}), \\ I_{x z} & = \sum_{i} m_{i} (- x_{i} z_{i}) . \\ ∴ L_{x} & = I_{x x} ω_{x} - x_{i}^{2} + I_{x y} ω_{y} + I_{x z} ω_{z} \end{aligned}

it can be asserted that:

\begin{aligned} L_{y} & = I_{y x} ω_{x} - x_{i}^{2} + I_{y y} ω_{y} + I_{y z} ω_{z}; \\ L_{z} & = I_{z x} ω_{x} - x_{i}^{2} + I_{z y} ω_{y} + I_{z z} ω_{z} \end{aligned}

define a matrix:

I = [\begin{matrix} I_{x x} & I_{x y} & I_{x z} \\ I_{y x} & I_{y y} & I_{y z} \\ I_{z x} & I_{z y} & I_{z z} \end{matrix}]

then:

\vec{L} = I \vec{ω}

eg: calculate the kinetic energy of rotation

\begin{aligned} T_{r o t} & = \frac{1}{2} \sum m_{i} {\vec{v}}_{i} \cdot {\vec{v}}_{i} \sim \frac{1}{2} m v^{2} \\ = \frac{1}{2} \sum m_{i} {\vec{v}}_{i} \cdot (\vec{ω} \times {\vec{r}}_{i}) \\ = \frac{1}{2} \sum m_{i} \vec{ω} \cdot ({\vec{r}}_{i} \times {\vec{v}}_{i}) \\ = \frac{1}{2} \vec{ω} \cdot \sum m_{i} ({\vec{r}}_{i} \times {\vec{v}}_{i}) \\ = \frac{1}{2} \vec{ω} \cdot \vec{L} \\ = \frac{1}{2} \vec{ω} \cdot I \cdot \vec{ω} \\ = \frac{1}{2} ω^{2} [\hat{n} \cdot I \hat{n}] \\ (1) & ∴ T_{r o t} & = \frac{1}{2} I ω^{2} \end{aligned}

comparing $(1)$ with $T = \frac{1}{2} m v^{2}$ , $I$ must have dimensions of $[M L^{2}] :$

T = \frac{1}{2} I ω^{2} = \frac{1}{2} I {\dot{θ}}^{2}

\begin{aligned} I & = \hat{n} \cdot \frac{1}{2} \vec{L} \\ \frac{1}{2} I ω^{2} & = \frac{1}{2} ω \hat{n} \cdot L \\ = \frac{1}{2} \vec{ω} \cdot \vec{L} \\ I & = \hat{n} \cdot \sum_{i} m_{i} (\hat{n} ({\vec{r}}_{i} \cdot {\vec{r}}_{i}) - {\vec{r}}_{i} ({\vec{r}}_{i} \cdot \hat{n})) \\ = \sum_{i} m_{i} [{\vec{r}}_{i} \cdot {\vec{r}}_{i} - ({\vec{r}}_{i} \cdot \hat{n})^{2}] \end{aligned}

continuum limit exists:

\sum_{i} m_{i} \to \sum_{i} ρ_{i} Δ V_{i} \to \int ρ (\vec{r}) d V

consider a cylinder having a length, $l$ , a radius, $r$ , and a constant density, $ρ$

\begin{aligned} I & = \int ρ (r) d V [\vec{r} \cdot \vec{r} - (\vec{r} \cdot \hat{z})^{2}] \\ = \sum_{i} m_{i} [(x^{2} + y^{2} + z^{2}) - z^{2}] \\ ∴ I & = \sum_{i} m_{i} {\tilde{r}}^{2} \end{aligned}

where, $\tilde{r} = x^{2} + y^{2}$

considering cylindrical coordinates: $d V = d x d y d z = \tilde{r} d \tilde{r} d θ d z$

\begin{aligned} I & = \int_{0}^{2} ρ \tilde{r} d \tilde{r} \int_{0}^{2 π} d θ \int_{0}^{l} d z {\tilde{r}}^{2} \\ ∴ I & = \frac{π l ρ R^{4}}{2} \\ ⟹ T_{r o t} & = \frac{1}{2} (\frac{π l ρ R^{4}}{2}) ω^{2} \end{aligned}

for a cylinder rotating through the centre of mass:

T_{t o t} = T_{r o t} + T_{t r a n s}