<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/7453ccbe-f81e-4ead-9971-cf103e932a47/Rigid_bodies.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/7453ccbe-f81e-4ead-9971-cf103e932a47/Rigid_bodies.png" width="40px" /> Rigid body: a system of particles in which the distance between them does not change regardless of the force acting.

</aside>

$$ \vec r^{(k)}=r_i^{(k)}\,\hat e_i $$

where $k=1\cdots N$ labels to the particles.

$$ 3 \;\text{translations} \quad ; \quad 3 \;\text{rotations} $$

$$ \begin{aligned} M&= \sum^N_{k=1}m_k=\text{total mass} \\ \vec R&= \frac 1M\sum^N_{k=1}m_k \vec r^{(k)}=\text{position vector of the CoM} \\ \vec P_\text{tot}&= \sum^N_{k=1}m_k\dot{\vec r}^{(k)}=\text{total linear momentum} \\ \vec L_\text{tot}&=\sum^N_{k=1}m_k\left (\vec r^{(k)}\times \dot{\vec r}^{(k)}\right )=\text{total angular momentum} \\ T_\text{tot}&=\frac 12 \sum^N_{k=1}m_k\left |\dot{\vec r}^{(k)}\right |^2=\text{total kinetic energy} \end{aligned} $$

These can all be turned into continuous limits as follows:

$$ \begin{aligned} M&= \int\rho\,\text dV \\ \vec R&= \frac 1M \int \rho\vec r\,\text dV \\ \vec P_\text{tot}&= \int \rho \dot{\vec r}\,\text dV\\ \vec L_\text{tot}&= \int \rho(\vec r \times \dot{\vec r})\,\text dV\\ T_\text{tot}&= \frac 12\int\rho|\dot{\vec r}|^2\,\text dV \end{aligned} $$

🧠 Remember: $\left . \dot{\vec r}^{(k)} \right |S=\left . \dot{\vec r}^{(k)} \right |{S'}+\vec \omega \times \vec r^{(k)}$

🧠 Remember: $\vec r^{(k)}=\vec R+{\vec r'}^{(k)}$

Moments of inertia

$$ I_{ij}=\int\text dM (r^2\delta_{ij}-r_ir_j)=\int\text dM\begin{bmatrix} y^2+z^2 & -xy & -xz \\ -yx & z^2+x^2 & -yz \\ -zx & -zy & x^2+y^2 \end{bmatrix} $$

Moment of inertia of a cylinder

Untitled

Consider a cylinder of radius $R$ and height $h$ about its center of mass

  1. If we have a solid cylinder with uniform density where

    $$ \text dM=\rho\,\text dV=\rho r\,\text dr\, \text d\theta \,\text dz \quad ; \quad \rho = \frac{M}{\pi R^2h} $$

    $$ \begin{aligned} I_{ij}&=\rho\int^R_0 r\,\text dr \int^{2\pi}0\text d\theta \int^{\frac 12 h}{- \frac 12 h}\text dz \begin{bmatrix} r^2\sin^2\theta+z^2 &-r^2\cos\theta\sin\theta & -rz\cos\theta \\ -r^2\cos\theta\sin\theta & r^2\cos^2\theta+z^2 &-rz\sin\theta \\ -rz\cos\theta & -rz\sin\theta & r^2

    \end{bmatrix} \\ &=\frac{M}{\pi R^2h}\int^R_0 r\,\text dr \begin{bmatrix} \pi r^2h+\frac 16 \pi h^3 & 0 & 0 \\ 0 & \pi r^2h+\frac 16 \pi h^3 & 0 \\ 0 & 0 & 2\pi r^2 h \end{bmatrix} \\ &=M\begin{bmatrix} \frac 14 R^2 + \frac 1 {12} h^2 & 0 & 0 \\ 0 & \frac 14 R^2 + \frac 1{12} h^2 & 0 \\ 0 & 0 & \frac 12 R^2 \end{bmatrix}

    \end{aligned} $$

  2. If we have a cylindrical shell with uniform density where

    $$ \text dM=\sigma \,\text dA=R\,\text d\theta \,\text dz \quad ;\quad \sigma =\frac{M}{2\pi Rh} $$

    $$ \begin{aligned} I_{ij}&=\sigma R\int^{2\pi}0\text d\theta \int^{\frac 12 h}{- \frac 12 h}\text dz \begin{bmatrix} R^2\sin^2\theta+z^2 &-R^2\cos\theta\sin\theta & -Rz\cos\theta \\ -R^2\cos\theta\sin\theta & R^2\cos^2\theta+z^2 &-Rz\sin\theta \\ -Rz\cos\theta & -Rz\sin\theta & R^2

    \end{bmatrix} \\ &=\frac{M}{2\pi h}\int^{\frac 12 h}_{- \frac 12 h}\text dz \begin{bmatrix} \pi R^2+2\pi z^2 & 0 & 0 \\ 0 & \pi R^2+2\pi z^2 & 0 \\ 0 & 0 & 2\pi R^2 \end{bmatrix} \\ &=M\begin{bmatrix} \frac 12 R^2+\frac 1{12} h^2 & 0 & 0 \\ 0 & \frac 12 R^2+\frac 1{12} h^2 & 0 \\ 0 & 0 & R^2 \end{bmatrix}

    \end{aligned} $$