Its not hard, it just unintuitive and depends on your understanding of circumference and angular momentum.
Analysis and solutions
First solution
If the smaller circle depends on the larger one (Case I), the larger circle's motion forces the smaller to traverse the larger’s circumference. If the larger circle depends on the smaller one (Case II), then the smaller circle's motion forces the larger circle to traverse the smaller circle’s circumference. This is the simplest solution.
Second solution
Well, both inner and outer wheel takes same time to travel from one point to another, so let it be T. as T is same, but circumferences are different, of course their linear velocities will be different. (Let r = radius of inner circle, R = radius of outer) so v(inner) = 2πr/T, V(outer) = 2πR/T where v < V. Now lets think of it in the angular perspective. We know v = rω^2. for cases of both wheels, as v and r are both varying, ω must stay constant for both as in an equation like (F = kx) if two variables are changing one variable is constant. so as ω is constant, and we previously mentioned T is constant so using θ = ω/T we can say angular displacement must remain same for both inner and outer wheel at all points in time. I think that's why both make a 360 degree rotation at the same despite inner circle having a smaller circumference.
Rotational speed, it's moving slower in the center.
many explanations have been put forward involving friction and slipping
Basic Physics explained it decades ago. Be better than this nonsense.
Yeah, that's why everyone gave a different answer.
Its not hard, it just unintuitive and depends on your understanding of circumference and angular momentum.
Analysis and solutions
First solution If the smaller circle depends on the larger one (Case I), the larger circle's motion forces the smaller to traverse the larger’s circumference. If the larger circle depends on the smaller one (Case II), then the smaller circle's motion forces the larger circle to traverse the smaller circle’s circumference. This is the simplest solution.
Second solution
Well, both inner and outer wheel takes same time to travel from one point to another, so let it be T. as T is same, but circumferences are different, of course their linear velocities will be different. (Let r = radius of inner circle, R = radius of outer) so v(inner) = 2πr/T, V(outer) = 2πR/T where v < V. Now lets think of it in the angular perspective. We know v = rω^2. for cases of both wheels, as v and r are both varying, ω must stay constant for both as in an equation like (F = kx) if two variables are changing one variable is constant. so as ω is constant, and we previously mentioned T is constant so using θ = ω/T we can say angular displacement must remain same for both inner and outer wheel at all points in time. I think that's why both make a 360 degree rotation at the same despite inner circle having a smaller circumference.