Physics

# Torque

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Note: for the purposes of this article, $\bold{ x }$ and $\overrightarrow{x}$ are equivalent notations for a column vector

## Introduction

Figures 1 and 2 depict a rod-shaped object accelerating due to a force $\overrightarrow{F}$ applied at its center of mass $m_c$

• For the cases depicted in Figures 1 and 2,
• The rod is (linearly) accelerating (represented by pink arrows) in the same direction as $\overrightarrow{F}$
• We can use $\overrightarrow{F} = m \overrightarrow{a}$ to find the acceleration of the object:

$\overrightarrow{a}_{\mathrm{object}} = \overrightarrow{F} / m_c$

Question: what happens when a force is applied to an object at a region that is not at the center of mass?

• Answer: assuming the object is free-floating in 3D space, it will rotate about the center of mass (see figure 3)
• Notes:
• $\overrightarrow{F}_\perp$ denotes the component of the force perpendicular to the object with respect to its center of mass
• Unlike cases 1 and 2, the resulting movement of the object (pink arrows) in case 3 is not in the same direction as $\overrightarrow{F}_\perp$

Now consider an object similar a hand on a clock (figures 4 and 5) where one component of this object is fixed in space (called the axis of rotation, pivot point, or fulcrum) and two other components (the arms) comprise a single rigid rod that is free to rotate about the pivot point (blue dot):

• Note: $\overrightarrow{r}$ denotes the moment arm distance or a vector whose origin starts at the pivot point and ends at the point of impact by a force $\overrightarrow{F}_\perp$ who direction is perpendicular to $\overrightarrow{r}$
• Applying a force at the pivot point wont cause the object to move (see case 4):
• BUT if we apply a force on a “clock-hand” component, it will accelerate in the direction that causes it to rotate about the pivot point (see case 5):

### Definitions

Torque (units in $\mathrm{Nm}$) describes the effect of $\overrightarrow{F}_\perp$ on an object with respect to the moment arm distance $\overrightarrow{r}$ at which a force is applied

• Equation:
• Let $F_\perp = \left| \overrightarrow{F}_\perp \right|$ and $r = \left| \overrightarrow{r} \right|$
• $\tau =\left| \overrightarrow{\tau} \right| = \overrightarrow{r} \times \overrightarrow{F}_\perp = r F_\perp$
• Note: sometimes, people use the term “moment” to describe torque
• Importantly, even though torque has units in $\mathrm{Nm}$ (same SI units for work ⇒ $1 \mathrm{J} = 1 \mathrm{Nm}$), we DON’T assign it units in joules
• Work ⇒ the subsequent change in an object’s position when a force applied is translational (i.e., non-rotating)
• Vectors describing  the force and object acceleration have the same direction (i.e. they are parallel)
• e.g., cases 1 and 2
• Torque ⇒ the subsequent change in an object’s position with respect to a pivot point when a force is applied  is rotational
• Vectors describing the force and moment arm distance are perpendicular relative to each other
• Vectors describing the force and object acceleration have different directions
• e.g., cases 3, 4, and 5

Torque is also be defined as the rate of change of angular momentum of an object

Torque convention:

• Positive torque ($\tau > 0$) ⇒ counterclockwise rotation
• Negative torque ($\tau < 0$) ⇒ clockwise rotation

## Sources

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