add uncommitted year 1 content

uni/mmme/1xxx/1028_statics_and_dynamics/dynamics.md

@@ -11,3 +11,62 @@ uuid: e6d3a307-b2e6-40e3-83bb-ef73512d69ad
 $$a_c = r\omega^2$$
 
 $$a = r\alpha \hat{e}_\theta - r\omega^2\hat{e}_r$$
+
+## Moment of Inertia
+
+$$J = mr^2 = \frac{M}{\ddot\theta}$$
+
+The unit of $J$ is kgm$^2$.
+
+Consider a particle of mass $m$ attached to one end of a rigid rod of length $r$.
+The rod is pivoting at its other end about point $O$, and experiences a torque $M$.
+This torque will cause the mass and the rod to rotate about $O$ with angular velocity
+$\dot{\theta}$ an angular acceleration $\ddot{\theta}$.
+
+![](./images/vimscrot-2022-03-10T14:40:59,716300890+00:00.png)
+
+What is the expression for $M$?
+
+Well if break down the moment $M$ into a force, $F$, acting on the mass, we know that the
+moment $M = Fr$.  
+We know $F = ma$, and this case $a = r\ddot{\theta}$ so $M = mr^2\ddot\theta$.
+
+The moment of inertia is $J = mr^2$ so $M = J\ddot\theta$.
+
+If multiple torques are applied to a body the *rotational equation* of the motion is
+
+$$\overrightarrow{M} = \sum_i \overrightarrow{M}_i = J \overrightarrow{\ddot\theta} = J \overrightarrow{\alpha}$$
+
+The moment of inertia of any object is found by considering the object to be made up of lots of
+small particles and adding the moments of inertia for each small particle.
+The moments of inertia for a body depends on the mass and its distribution about the axis in
+consideration.
+
+$$J = \sum_i m_ir^2_i \rightarrow \int\! r^2 \mathrm{d}m$$
+
+### Perpendicular Axis Rule
+
+The perpendicular axis rule states that, for lamina object:
+
+$$J_z = J_x + J_y$$
+
+where $J_x$, $J_y$, and $J_z$ are the moments of inertia along their respective axes.
+
+### Parallel Axes Rule (Huygens-Steiner Theorem)
+
+The parallel axes rule states that:
+
+$$J_A = J_G = md^2$$
+
+where $d$ is the perpendicular distance between the two axes.
+
+![](./images/vimscrot-2022-03-10T15:06:48,355133323+00:00.png)
+
+### Moment of a Compound Object
+
+The moment of inertia for any compound object can be calculated by adding and subtracting the
+moments of inertia for its 'standard' components.
+
+### Moment of Inertia of Standard Objects
+
+// TODO