# Angular displacement

**Angular displacement** of a body is the angle in radians (degrees, revolutions) through which a point or line has been rotated in a specified sense about a specified axis. When an object rotates about its axis, the motion cannot simply be analyzed as a particle, since in circular motion it undergoes a changing velocity and acceleration at any time (*t*). When dealing with the rotation of an object, it becomes simpler to consider the body itself rigid. A body is generally considered rigid when the separations between all the particles remains constant throughout the objects motion, so for example parts of its mass are not flying off. In a realistic sense, all things can be deformable, however this impact is minimal and negligible. Thus the rotation of a rigid body over a fixed axis is referred to as rotational motion.

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## Example

In the example illustrated to the right, a particle on object P at a fixed distance *r* from the origin, *O*, rotating counterclockwise. It becomes important to then represent the position of particle P in terms of its polar coordinates (*r*, *θ*). In this particular example, the value of *θ* is changing, while the value of the radius remains the same. (In rectangular coordinates (*x*, *y*) both *x* and *y* vary with time). As the particle moves along the circle, it travels an arc length *s*, which becomes related to the angular position through the relationship:

## Measurements of angular displacement

Angular displacement may be measured in radians or degrees. If using radians, it provides a very simple relationship between distance traveled around the circle and the distance *r* from the centre.

For example if an object rotates 360 degrees around a circle of radius *r*, the angular displacement is given by the distance traveled around the circumference - which is 2π*r*
divided by the radius: which easily simplifies to . Therefore 1 revolution is radians.

When object travels from point P to point Q, as it does in the illustration to the left, over the radius of the circle goes around a change in angle. which equals the **Angular Displacement**.

## Three dimensions

In three dimensions, angular displacement is an entity with a direction and a magnitude. The direction specifies the axis of rotation, which always exists by virtue of the Euler's rotation theorem; the magnitude specifies the rotation in radians about that axis (using the right-hand rule to determine direction).

Despite having direction and magnitude, angular displacement is not a vector because it does not obey the commutative law for addition.^{[1]}

### Matrix notation

Given that any frame in the space can be described by a rotation matrix, the displacement among them can also be described by a rotation matrix. Being and two matrices, the angular displacement matrix between them can be obtained as

## References

- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}