diff --git a/docs/ce1/ece105.md b/docs/ce1/ece105.md index 613c168..bf17244 100644 --- a/docs/ce1/ece105.md +++ b/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$$