> [!summary] Torque is the force that causes an object to rotate > **Key Equations:** > Torque: $\tau = \vec{r}\times \vec{F}$ >[!info]+ Read Time **⏱ 2 mins** # Definition Torque is the twisting force that enables an object to rotate. This is always needed for an object to experience a force to begin or continue rotation. It is defined as the [[Derivative|derivative]] of [[Angular Momentum|angular momentum]], which physically is the amount of twisting force at a point. ## Derivation > [!warning] Assumptions To derive torque, assume the following: > - [[Angular Momentum|Angular momentum]] is defined as $\vec{L} = \vec{r} \times \vec{p}$ > - The [[Derivative|derivative]] of angular momentum is torque $\frac{dL}{dt}=\tau$ $\begin{array}{c} \vec{L}=\vec{r}\times \vec{p} \\ \frac{d\vec{L}}{dt} = \frac{d\vec{r}}{dt} \times \vec{p} + \vec{r} \times \frac{d\vec{p}}{dt} \\ \frac{d\vec{L}}{dt} = \cancel{ m(\vec{v}\times \vec{v})_{} }+ \vec{r} \times \vec{F} \\ \frac{d\vec{L}}{dt} = \vec{r} \times \vec{F} \\ \tau = \vec{r}\times \vec{F} \end{array}$ > [!note] So by definition torque has the twisting force when the force applied is perpendicular to the r vector and/or the r vector is the farthest possible distance away from the reference point. > [!note]+ Torque Diagram > ![[tor_1.png]] > ![[tor_4.png]] ## Deriving Torque in Term of Moment of Inertia >[!warning] Assumptions From the definition of torque, torque has the most [[Energy|energy]] when it [[Tangential & Perpendicular|perpendicular]] to the r [[Scalar & Vectors|vector]]. So to derive an equation of that assume the following: > - Torque with a [[Tangential & Perpendicular|perpendicular]] force is defined as $\tau=\vec{r} \times \vec{F_{\perp}=\vec{r}F_{\perp}\sin(90)}$ > - Since the force is perpendicular by definition of [[Newton Laws|newtons second law]] and [[Angular Acceleration|angular acceleration]] $F_{\perp}=ma_{t}=mr\alpha$ > - The [[Moment of Inertia|moment of inertia]] is defined as $I=mr^2$ $\begin{align*} \tau &= rF_\perp \sin(90) \\ &= mr^2 \alpha \\ \tau &= I\alpha \end{align*}$ --- > 🧠 Enjoy this walkthrough? [Support Math & Matter](https://github.com/rajeevphysics/Obsidian-MathMatter) with a star and help others learn more easily. ---