Torque

Summary

Torque is the force that causes an object to rotate

Key Equations:

Torque:
$τ = \vec{r} \times \vec{F}$

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 of angular momentum, which physically is the amount of twisting force at a point.

Derivation

Assumptions

To derive torque, assume the following:

Angular momentum is defined as $\vec{L} = \vec{r} \times \vec{p}$
The derivative of angular momentum is torque $\frac{d L}{d t} = τ$

\begin{matrix} \vec{L} = \vec{r} \times \vec{p} \\ \frac{d \vec{L}}{d t} = \frac{d \vec{r}}{d t} \times \vec{p} + \vec{r} \times \frac{d \vec{p}}{d t} \\ \frac{d \vec{L}}{d t} = m (\vec{v} \times \vec{v})_{} + \vec{r} \times \vec{F} \\ \frac{d \vec{L}}{d t} = \vec{r} \times \vec{F} \\ τ = \vec{r} \times \vec{F} \end{matrix}

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.

Torque Diagram

!tor_1.png
!tor_4.png

Deriving Torque in Term of Moment of Inertia

Assumptions

From the definition of torque, torque has the most energy when it perpendicular to the r vector. So to derive an equation of that assume the following:

Torque with a perpendicular force is defined as $τ = \vec{r} \times \vec{F_{⊥} = \vec{r} F_{⊥} \sin (90)}$
Since the force is perpendicular by definition of newtons second law and angular acceleration $F_{⊥} = m a_{t} = m r α$
The moment of inertia is defined as $I = m r^{2}$

\begin{aligned} τ & = r F_{⊥} \sin (90) \\ = m r^{2} α \\ τ & = I α \end{aligned}

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