1 Rotational Variables
Rotation correspond to the motion of an object about a
The proper study of this type of motion requires several
•Deﬁne a good reference system. In the ﬁgure, zaxis
coincides with the rotation axis.
•Deﬁne angular variables that simplify the math for
describing the position of the object respect the ref-
Def. 1 (Rigid Body).A rigid body is a particle system where
the distances between all of them remain ﬁxed at all times. In
other words, the body does not suffer deformations.
A rigid body is sometimes called also solid.
Why we concentrate on the study of rigid bodies? Because
we can study the motion of the complete body paying at-
tention to a single point of the object, and then refer the
rest of the body to that point.
A very useful reference system selection is one where one
of the axis coincides with the rotation axis.
On that case, we can describe the position of each part of
the object using the angle between a given part of the ob-
ject and an arbitrary axis x.
Def. 2 (Angular Position).The angular position is the angle
of the reference line relative to a ﬁxed direction (xaxis), which
we take as the zero angular position.
The angular position is denoted by the letter θand its measured
in the counterclockwise direction.
where sis the arc length deﬁned by the reference point and ris
the distance from this point to the rotation axis, which coincides
with the origin of the reference system.
Angles are measured in radians ([rad]) instead of degrees
or revolutions. A complete revolution describes an arc of
s= 2πr so the angle is
Def. 3 (Angular Displacement).The angular displacement
correspond to the angle difference measured at two different
times. It is represented by ∆θand its value is
Obs. 1. Angle displacements can be positive or negative de-
pending on the direction of the rotation.
•Positive: the body rotates counterclockwise.
•Negative: the body rotates clockwise.
Def. 4 (Angular Velocity).Let θ1be the angular position at
t=t1and θ2be the angular position at t=t2.
The average angular velocity on the interval [t1, t2]is