diff --git a/docs/ce1/ece105.md b/docs/ce1/ece105.md index 16bfb48..ce0174c 100644 --- a/docs/ce1/ece105.md +++ b/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. + +(Source: Wikimedia Commons) + +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: + +$$ +v_{cm} = \sqrt{\frac{2}{1+c}gh} \\ +a_{cm} = \frac{g\sin\theta}{1+c} +$$