The Whirlygig.Designers of amusement park rides need to know a lot of physics. Consider those rides that carry passengers in dizzying complex motions, subjecting them to gut wrenching twists and turns with changes of velocity and acceleration.
Here's a design for such a ride, sketched by an apprentice employee. He called it the "whirlygig". It has four arms attached to a rotating hub. The hub is shown in blue. At the ends of the four arms are rotating wheels carrying passengers (yellow). Each yellow wheel carries four pasengers, so the ride accomodates sixteen. Only two passengers are shown as black circles at C and D. To complicate the motion, our clever designer has a pulley fixed to the center axle (green), which does not rotate. A similar pulley of the same size is shown fixed to the upper yellow wheel at B. (It rotates with that wheel.) There's a belt drive connecting these two pulleys. Similar pulleys drive the other yellow wheels in exactly the same way. These have been omittited from the diagram to avoid complications and confusion.
The motions depicted in the diagram take place in a horizontal plane.
The rotation of the center hub drives the four arms and the yellow wheels. But the central green pulley does not rotate.
When do passengers C and D come closest together. When farthest apart?
What is the speed of each passenger as a function of time?
What is the acceleration of each passenger as a function of time?
What is the geometric shape of the path of each passenger?
Look for simple methods to obtain the solution.
Answers.At first look, you might assume that the motion of each passenger is quite complex, some fancy cycloid, perhaps. Actually the paths are simple. Each passenger moves in a circle at constant speed. All these circles have the same radius and a fixed center, but all are separate. So the acceleration of each is a vector of constant size pointed toward the center of that circle. A simple merry-go-round would give the same experience.
And, perhaps surprisingly, the distance between any two passengers, C and D, for example, is constant throught the motion.
If the center hub and arms rotate with angular speed counter-clockwise, the four yellow wheels rotate at the same angular speed, also counter-clockwise. So if a passenger sits on a yellow wheel facing, say, North, he will remain facing North during the entire ride. Not unlike a Ferris wheel ride, where passengers face the same direction during the entire revolution.
It's unlikely this particular design would achieve popularity in amusement parks. Similar amuseument park rides drive the outer wheels differently, often in the opposite direction compared to our failed design.
Footnote: This puzzle was concocted after I reviewed a paper submitted to a journal, in which the author misunderstood the earth's motion about the earth-moon barycenter and failed to realize that the centripetal forces due to that motion cannot raise ocean tidal bulges. I find it fascinating how apparently unrelated situations often have underlying common physics. Looking for such similarities can strengthen one's physical insight. Educators call this knowledge transference.