ece105: add angular momentum and torque

docs/ce1/ece105.md

@@ -91,7 +91,7 @@ $$I=\int^M_0 R^2 dm$$
 - 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: $I=\frac{1}{3}ML^2$
+- Thin rectangular plate about perpendicular axis through center: $I=\frac{1}{3}ML^2$
 
 ### Rotational-translational equivalence
 
@@ -107,7 +107,7 @@ Angular velocity is related to velocity:
 
 $$\omega = \frac{v}{r}$$
 
-The direction of the tangential values can be determined via the right hand rule.
+The direction of the tangential values can be determined via the right hand rule. Where $r$ is the vector from the **origin to the mass**:
 
 $$
 \vec v = r\times\omega \\
@@ -124,3 +124,32 @@ And all kinematic equations have their rotational equivalents.
 Most translational equations also have rotational equivalents.
 
 $$E_\text{k rot} = \frac{1}{2}I\omega^2$$
+
+## Torque
+
+Torque is the rotational equivalent of force.
+
+$$\vec\tau=I\vec\alpha$$
+$$\vec\tau=\vec r\times\vec F$$
+$$|\vec\tau=|r||F|\sin\theta$$
+
+In the general case, especially when the force is variable, the work done is equal to the integral of force over displacement.
+
+$$W=\int^{x_f}_{x_i}F_xdx$$
+
+Work is also related to torque:
+
+$$W=\tau\Delta\theta$$
+$$W=F\Delta S$$
+
+The total net work from torque from external forces is equivalent to:
+
+$$W=\Delta E_k = \int^{\theta_f}_{\theta_i}\taud\theta$$
+
+### Angular momentum
+
+This is the same as linear momentum.
+
+$$\vec L = \vec r\times\vec p$$
+$$\vec L = I\vec\omega$$
+$$\vec L =\vec\tau t$$