Calculate linear velicity at a point in a rotating object If I have an object $A$ rotating at an angular velocity $w$, how would I calculate the linear velocity $v(x, y)$ of a certain point $r(x, y)$ in the object?
 A: Think of it like this - if the entire object is rotating with angular velocity $\omega = 2\pi/T$ (presumably about the origin), that means each point on the object travels a distance $2\pi R$ in a time $T$, where $R$ is the distance from the origin to that point. The speed of the object is of course constant, and therefore the linear speed at any point on the object will be $v = 2\pi R / T = R\omega$. Now we simply plug in $R = \sqrt{x^2+y^2}$ and get
$$v(x,y) = \omega R(x,y) = \omega \sqrt{x^2+y^2}$$
A: Consider the center of rotation, point C, and some arbitrary point A with coordinates $(x,y)$. If the rotational speed is $\Omega$ then point A is going to orbit around C.

The velocity of point A is a vector perpendicular to CA and by analyzing the similar triangles they form (look at the angle $\theta$) you find the velocity vector of A to be
$$ v_A = \pmatrix{-y\, \Omega \\ x\, \Omega} $$
The speed of A is $$ \| v_A \| = \sqrt{ (-y \Omega)^2 + (x \Omega)^2} = \Omega \sqrt{x^2+y^2}$$
which is the product of the rotational speed $\Omega$ and the distance to A.
