[latexpage]

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

## Introduction

Figures 1 and 2 depict a rod-shaped object accelerating due to a force applied at its center of mass

- For the cases depicted in Figures 1 and 2,
- The rod is (linearly) accelerating (represented by pink arrows) in the same direction as
- We can use to find the acceleration of the object:

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:
- 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

- Notes:

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: 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 who direction is perpendicular to - 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 ) describes the effect of on an object with respect to the moment arm distance at which a force is applied

- Equation:
- Let and

- Note: sometimes, people use the term “moment” to describe torque
- Importantly, even though torque has units in (same SI units for work ⇒ ), 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

- Work ⇒ the subsequent change in an object’s position when a force applied is

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

Torque convention:

- Positive torque () ⇒ counterclockwise rotation
- Negative torque () ⇒ clockwise rotation

## Sources

- https://www.khanacademy.org/science/physics/torque-angular-momentum/modal/v/introduction-to-torque
- https://en.wikipedia.org/wiki/Torque