Mechanics 1: Linear Kinematics and Calculus - 3D Math Primer for Graphics and Game Development - page 549

essentially discards any velocity parallel to r ; only velocity perpendicular
to r contributes to the results.
If the particle p is indeed orbiting o at constant speed, then the an-
gular velocity computed by Equation (11.32) will be constant. In general,
however, the angular velocity measured relative to any arbitrary point is
not constant. For example, consider a particle moving with constant linear
velocity. The angular velocity measured relative to a stationary point o
will grow as the particle approaches o , reaches a maximum at the point
of closest approach, and then decreases. Furthermore, even if the particle
is moving in an orbital path, the angular velocity will be a constant only
when measured relative to the center of the orbit.
One extremely important example of orbital motion in 3D is a particle
attached to a rigid body rotating about an axis. Let's choose o to be at
the intersection of the axis of rotation and the plane that contains the cir-
cular orbit of p ; This causes r to be perpendicular to the axis of rotation.
Under these assumptions, the orbital angular velocity computed by Equa-
tion (11.32) is the same for every particle, and it's also the same as the spin
angular velocity of the rigid body. We have more to say about this in the
next chapter.
We don't often need to calculate angular velocity of a point relative to
some point that isn't the center of the orbit. (However, Equation (11.31)
is used frequently to compute a linear point velocity based on its orbital
velocity.) So why do we talk about this? Because the computation is similar
to the way we measure torque (see Section 12.5) for a force applied at an
arbitrary direction at an arbitrary location.

11.9 Exercises

(Answers on page 781.)

1. The Pascal is a unit of measurement for pressure, defined as one Newton
per square meter. One Pascal is equal to how many psi? (The psi is one
pound of force per square inch.)

2. The 1D position of a particle is described piecewise by

<
:

2t−t 2 0 ≤ t < 2,

0 2 ≤ t < 4,

x(t) =

sin(πt) 4 ≤ t < 7,

7−t 7 ≤ t.

Plot a graph of the particle's motion.