Mechanics 1: Linear Kinematics and Calculus - 3D Math Primer for Graphics and Game Development - page 546

Differentiating once more to get the acceleration, we have

x(t) = d

Acceleration as a
function of time

dt (−rω sin(θ 0 + ωt)) = −rω 2 cos(θ 0 + ωt),
y(t) = d

dt (rω cos(θ 0 + ωt)) = −rω 2 sin(θ 0 + ωt).

These results agree with our earlier findings. Comparing the acceleration
functions to the position, we confirm that they do indeed point in opposite
directions. Furthermore, recalling that ω = s/r, we note that, as predicted,
acceleration has a length of s 2 /r.
Sometimes ω is more immediately accessible than s. In these situations,
it's useful to be able to express the magnitude of the centripetal acceleration
just in terms of ω and r. Solving ω = s/r for s gives us s = rω. Plugging
this in to Equation (11.29), we have

Acceleration in terms of
angular speed ω and
radius r

a = s 2 /r = (rω) 2 /r = rω 2 . (11.30)

Let's work through an interesting example, the results of which will be
useful in later sections. All of us are aboard a spinning centrifuge right
now: Earth! Earth's rotation creates an apparent centrifugal force, which
tends to throw us away from the Earth's center. Luckily, Earth's gravity is
strong enough to keep us here. Given that Earth's mean radius is 6,371 km,
what is the centripetal acceleration experienced at the equator?
To answer this question, we use Equation (11.30). The radius was given
as r = 6,371 km, and the rotation rate is ω = 2π/day.

Centripetal acceleration
at the equator due to
Earth's rotation

a = rω 2 = (6 371 km)(2π/day) 2 = (6.371 × 10 6 m)(2π/(86 400 s)) 2
≈ (6.371 × 10 6 m)(5.2885 × 10

−9 s −2 ) ≈ 0.03369 m/s 2 .

What about the magnitude of the centripetal acceleration at the poles? Is
it the same? Keep this question in mind; we return to it in Section 12.2.1.

11.8.2 Uniform Circular Motion in Three Dimensions
So far, we've essentially been working in two dimensions, operating “in the
plane” and not concerning ourselves with how this plane might be oriented
in three dimensions. Now let us consider the more general case. We wish
to describe the position, velocity, and acceleration of the particle as three-
dimensional vectors, where the axis of rotation (which is perpendicular to
the plane containing the circular path) is arbitrarily oriented.
Suppose a particle at position p is moving in a circular path around
point o . Since there are many different circular paths that contain both o
and p , we must also specify an axis of rotation perpendicular to the plane.