Torque and Rotational Mechanics

Core Definition

Torque ( $\vec{\tau}$ ) (also known as the moment of force) is a vector quantity that measures the tendency of a force to rotate an object about a pivot point or axis:

\vec{\tau} = \vec{r} \times \vec{F}

where:

$\vec{r}$ is the position vector from the pivot point (origin) to the point where the force is applied (in meters, $\text{m}$ ).
$\vec{F}$ is the applied force vector (in Newtons, $\text{N}$ ).
$\vec{\tau}$ is the resulting torque vector, measured in Newton-meters ( $\text{N}\cdot\text{m}$ ).

Concept Explanation

Because torque is defined via the Cross Product, it is strictly a vector quantity:

Magnitude of Torque:
$|\vec{\tau}| = |\vec{r}| |\vec{F}| \sin\theta = |\vec{r}_{\perp}| |\vec{F}|$
where $\theta$ is the angle between $\vec{r}$ and $\vec{F}$ , and $|\vec{r}_{\perp}| = |\vec{r}|\sin\theta$ is the effective lever arm.
- Maximum torque occurs when the force is applied perpendicular to the lever arm ( $\theta = 90^\circ, \sin 90^\circ = 1$ ).
- Zero torque occurs when the force points directly toward or away from the pivot ( $\theta = 0^\circ$ or $180^\circ$ ).
Direction of Torque (Axis of Rotation): The torque vector points perpendicular to both the position vector and the force vector according to the Right-Hand Rule. The direction of $\vec{\tau}$ defines the physical axis of rotation (counterclockwise positive according to right-hand thumb orientation).

Physical Units Distinction: Work vs. Torque

Mechanical Work ( $W = \vec{F} \cdot \vec{d}$ ): Scalar energy measured in Joules ( $\text{J} = \text{N}\cdot\text{m}$ ).
Torque ( $\vec{\tau} = \vec{r} \times \vec{F}$ ): Vector rotational moment measured in $\text{N}\cdot\text{m}$ (never called Joules, because it represents moment of turning effort, not scalar energy transfer).

Engineering Applications

Robotics & Mechatronics: Sizing joint actuator motors to counteract gravitational torque and dynamic payload moments on robotic arms.
Structural Engineering: Calculating bending moments at support columns and cantilevered beams.
Automotive Engineering: Engine crankshaft torque transmission and fastener bolt tightening specifications.

Example Problems

Example 1: 3D Torque on a Robot Joint

A robotic arm holds a tool at $\vec{r} = \langle 0.8, 0, 0.6 \rangle\text{ m}$ from its shoulder pivot. A vertical load $\vec{F} = \langle 0, -100, 0 \rangle\text{ N}$ acts on the tool. Find the torque vector $\vec{\tau}$ and its magnitude.

Solution

Compute Cross Product via Determinant: $\vec{\tau} = \vec{r} \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0.8 & 0 & 0.6 \\ 0 & -100 & 0 \end{vmatrix}$
Expand Along Top Row: $\vec{\tau} = \hat{i}(0\cdot 0 - 0.6(-100)) - \hat{j}(0.8\cdot 0 - 0.6\cdot 0) + \hat{k}(0.8(-100) - 0\cdot 0)$ $\vec{\tau} = \hat{i}(60) - \hat{j}(0) + \hat{k}(-80) = \langle 60, 0, -80 \rangle\text{ N}\cdot\text{m}$
Compute Torque Magnitude: $|\vec{\tau}| = \sqrt{60^2 + 0^2 + (-80)^2} = \sqrt{3600 + 6400} = \sqrt{10000} = 100\text{ N}\cdot\text{m}$

Connections & References

Parent Concepts: Cross Product | Dot and Cross Products
Related Notes: Dot Product | Mechanical Work
Course Sessions: Calc-3 Session 03