EDIT: After sitting on this and digesting a bit with the comments, I realized I really want both the maximum and minimum speeds. The selected issue should get me there.
I'm trying to calculate the effective linear speed of the dough hook arm of a stand mixer, such as a KitchenAid mixer. If I can figure out the arc length I think I can get the speed easily enough. I also suspect that polar coordinates would help solve this, but it has been way too long since I've had to use polar coordinates and I can't even remember how to set this up.
Below is a quick schematic of how I'm looking at this problem. We have a dough hook attachment which is basically a stick on an “arm” of length "r" (70 mm in my case). This arm is then attached to a larger “ARM” with radius "R". ARM rotates at a given speed (approximately 45 revolutions per minute). For each complete revolution of ARM, arm rotates X times (say 2.5 times). If I am looking at point B, for one complete revolution of ARM, how far has B traveled through space? Or how fast is B moving through space, but if I can get the distance, I can use RPM to get speed easily enough.
Thank you all for the help! Let me know if you need more information or if I asked this in the wrong way. If I missed an answer that already provided this information, chances are I just didn't understand the terminology well enough to realize it was the same question.
TLDR: See schematic below, point “B” is orbiting around another revolving point “A”. A orbits around a fixed point in space with radius “R”, and B orbits around A with a radius of “r”. For each complete revolution of A, B completes “X” orbits. How do I calculate the arc length (and speed) of B with respect to the center of A’s orbit?
For example numbers assume: R = 250 mm, r = 50 mm, X = 2.5
A rough schematic of the dough hook traveling around the bowl of the mixer, with the circular arc of point A in the center, and two possible arcs for point B on the right.
The velocity vector of point $A$ relative to axis over one period is $$(R\omega_R\cos(\omega_Rt),R\omega_R\sin(\omega_Rt)).$$ Now, for simplicity assume that points $A$ and $B$ start their travels with the same phase. Then, the velocity of point $B$ relative to point $A$ is $$(r\omega_r\cos(\omega_rt),r\omega_r\sin(\omega_rt)).$$ Then, total velocity vector of point $B$ relative to axis of mixer is $$v = (R\omega_R\cos(\omega_Rt)+r\omega_r\cos(\omega_rt),R\omega_R\sin(\omega_Rt)+r\omega_r\sin(\omega_rt)).$$ Speed at any time $t$ is given by $$s(t) = \sqrt{(R\omega_R\cos(\omega_Rt)+r\omega_r\cos(\omega_rt))^2+(R\omega_R\sin(\omega_Rt)+r\omega_r\sin(\omega_rt))^2}.$$
Total distance travelled over one cycle is $$D = \int_{0}^{T} s(t)dt.$$
Formula for $\omega_R$: Time how long it takes for point $A$ to make one revolution, say $T_A$. Then, $\omega_R = 2\pi/T_A$.
Formula for $\omega_r$: Count how many loops during rotation, say $n$. Then, $\omega_r = n\omega_R$.
Finally, if you are trying to calculate "effective" linear speed, I assume that you want and average speed in some sense. One way to do this is simply compute $s_{eff} = D/T$ using formulas above.