ece105: add moment and rotational equivalence

docs/ce1/ece105.md

@@ -61,4 +61,66 @@ $$
 $$
 
 This means that a hole in a rod can use a different formula:
-$$x_{cm} = \frac{1}{M}\int^M_0 xdm$$
+$$x_{cm} = \frac{1}{M}\int^M_0 x\cdot dm$$
+
+For a solid object, the centre of mass can be expressed as a Riemann sum and thus an integral:
+
+$$r_{cm} = \frac{1}{M}\int_0^M r\cdot dm$$
+
+In an **isolated system**, it is guaranteed that the centre of mass of the whole system never changes so long as only rigid bodies are involved.
+
+## Rotational motion
+
+### Moment of inertia
+
+The moment of inertia of an object represents its ability to resist rotation, effectively the rotational equivalent of mass. It is equal to the sum of each point and distance from the axis of rotation.
+
+$$I=\sum(mr)^2$$
+
+For more complex objects where the distance often changes:
+
+$$I=\int^M_0 R^2 dm$$
+
+#### Common moment shapes
+
+- Solid cylinder or disc symmetrical to axis: $I = \frac{1}{2}MR^2$
+- Hoop about symmetrical axis: $I=MR^2$
+- Solid sphere: $\frac{2}{5}MR^2$
+- Thin spherical shell: $I=\frac{2}{3}MR^2$
+- Solid cylinder about the central diameter: $I=\frac{1}{4}MR^2 + \frac{1}{12}ML^2$
+- Hoop about diameter: $I=\frac{1}{2}MR^2$
+- Rod about center: $I=\frac{1}{12}ML^2$
+- Rod about end: $I=\frac{1}{3}ML^2$
+- Square slab about perpendicular axis through center: $I=\frac{1}{3}ML^2$
+
+### Rotational-translational equivalence
+
+Most translational variables have a rotational equivalent.
+
+Although the below should be represented as cross products, this course only deals with rotation perpendicular to the axis, so the following are always true.
+
+Angular acceleration is related to acceleration:
+
+$$\alpha = \frac{a}{r}$$
+
+Angular velocity is related to velocity:
+
+$$\omega = \frac{v}{r}$$
+
+The direction of the tangential values can be determined via the right hand rule.
+
+$$
+\vec v = r\times\omega \\
+\vec a = r\times\alpha
+$$
+
+And all kinematic equations have their rotational equivalents.
+
+- $\theta = \frac{1}{2}(\omega_f + \omega_i)t$
+- $\omega_f = \omega_i + \alpha t$
+- $\theta = \omega_i t + \frac{1}{2}\alpha t^2$
+- $\omega_f^2 + \omega_i^2 + 2\alpha\theta$
+
+Most translational equations also have rotational equivalents.
+
+$$E_\text{k rot} = \frac{1}{2}I\omega^2$$