Let L be the circle in the x-y plane with center the origin and radius 38.
Let S be a moveable circle with radius 22 . S is rolled
along the inside of L without slipping while L remains fixed.
A point P is marked on S before S is rolled and the path of P is studied.
The initial position of P is (38,0).
The initial position of the center of S is (16,0) .
After S has moved counterclockwise about the origin
through an angle t the position of P is
$$x= 16 \cos t + 22 \cos \left( \frac{8}{11} t \right)$$ $$y= 16 \sin t - 22 \sin \left( \frac{8}{11} t \right)$$ How far does P move before it returns to its initial position?
Hint: You may use the formulas for cos( u+v) and sin( w /2).
S makes several complete revolutions about the origin before P returns to (38,0).