Two quarters are placed on a table so that they are touching. One of them is hel

Question

Two quarters are placed on a table so that they are touching. One of them is held in a fixed position, and the other is rolled around its circumference. How many complete revolutions has the rolling quarter made, after it returns to its starting position? (epicycloid) An epicycloid is the curve traced out by a point on a circle that rolls around the circumference of a fixed circle. It is assumed that friction determines the angular velocity of the rolling circle. If the fixed circle has radius R and the rolling circle has radius r, there exists a parametrization of the form X (t) = {{R + r) cos t, (R + r) sin t) - (r cos wt, r sin wt), where w is the angular velocity of the rolling circle. What is w, in terms of R and r? Use a computer to generate pictures of three examples of epicycloids with R = 1, R = 7, and R = 7/5, with r = 1 in each case. Assuming that R greaterthanorequalto r, make a conjecture about the number of cusps (i.e. places where the point touches the fixed circle), as a function of r and R, and prove it. In cases where R is not a multiple of r, you should assume that the circle rolls around enough times to make every possible point of contact.

Explanation / Answer

Answering first question

The Rolling coin will make exactly 2 revolution before coming to initial point as fixed will rotate in back direction.Think of it as both quarters turning at the same speed but in opposite directions, except the observer rotates around one of the quarters. Instead of each turning once, one turns zero and one turns twice

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