Game Development Reference
In-Depth Information
Figure 2-11. Tangential and centripetal acceleration
The formula for the magnitude of centripetal acceleration, a n , is:
a n = v 2 /r
where v is the tangential velocity. Substituting the equation for tangential velocity into
this equation for centripetal acceleration gives the following alternative form:
a n = r ω 2
In two dimensions you can easily get away with using these scalar equations; however,
in three dimensions you'll have to use the vector forms of these equations. Angular
velocity as a vector is parallel with the axis of rotation. In Figure 2-10 the angular velocity
would be pointing out of the page directly at you. Its sense, or direction of rotation, is
determined by the right hand rule . If you curl the fingers of your right hand in an arc
around the axis of rotation with your fingers pointing toward the direction in which
the body is rotating, then your thumb will stick up in the direction of the angular velocity
vector.
If you take the vector cross product (refer to the sidebar “Vector Cross Product” on page
67 for background and to Appendix A for a review of vector math) of the angular velocity
vector and the vector from the axis of rotation to the particle under consideration, you'll
end up with the linear, tangential velocity vector. This is written as follows:
v = ω × r