ece105: add rolling motion

docs/ce1/ece105.md

@@ -153,3 +153,44 @@ This is the same as linear momentum.
 $$\vec L = \vec r\times\vec p$$
 $$\vec L = I\vec\omega$$
 $$\vec L =\vec\tau t$$
+
+## Rolling motion
+
+!!! definition
+    - **Slipping** is spinning without sliding.
+    - **Skidding** is sliding without spinning.
+
+Pure rolling motion is **only true if** the tangential velocity of the centre of mass is equal to its rotational velocity.
+
+$$v_{cm}=R\omega$$
+
+In pure rolling motion, the point at the top is moving at two times the velocity while the point at the bottom has no tangential velocity.
+
+<img src="https://upload.wikimedia.org/wikipedia/commons/8/8d/Velocitats_Roda.svg" width=500>(Source: Wikimedia Commons)</img>
+
+For any point in the mass:
+
+$$
+v_{net} = v_{trans} + v_{rot} \\
+v_{net} = v_{cm} + \vec R \times\vec\omega \\
+E_{k roll} = E_{k trans} + E_{k rot}
+$$
+
+Alternatively, the rolling can be considered as a rotation about the pivot point between the disk and the ground, allowing rolling motion to be represented as rotational motion around the pivot point. The **parallel axis theorem** can be used to return it back to the original point.
+
+$$\sum\tau_b=I_b\alpha$$
+
+At least one external torque and one external force is required to initiate pure rolling motion because the two components are separate.
+
+If an object is purely rolling and then it moves to:
+
+- a flat, frictionless surface, it continues purely rolling
+- an inclined, frictionless surface, external torque is needed to maintain pure rolling motion
+- an inclined surface with friction, it continues purely rolling
+
+Where $c$ is the coefficient to the moment of inertia ($I=cMR^2$), while rolling down an incline:
+
+$$
+v_{cm} = \sqrt{\frac{2}{1+c}gh} \\
+a_{cm} = \frac{g\sin\theta}{1+c}
+$$