Cylinder Rolling inside Another Rolling Cylinder

Discuss the motion of a cylinder that rolls without slipping inside another cylinder, when the latter rolls without slipping on a horizontal plane. 2 Solution This problem is a variant of the case of one cylinder rolling on the outside of another rolling cylinder [1]. Special cases involving cylindrical shells or a solid inner cylinder are considered in ex. 2, p. 372 of [2], and in sec. 8.5, p. 111 of [3]. When one cylinder is directly above the other, we define the line of contact of the outer cylinder, 1, with the horizontal plane to be the z-axis, at x = y = 0. Then, the condition of rolling without slipping for the outer cylinder, of outer radius R1 is that when it has rolled (positive) distance x1, the initial line of contact has rotated through angle φ1 = x1/R1, clockwise with respect to the vertical, as shown in the figure below. This rolling constraint can be written as x1 = R1φ1. (1) Meanwhile, if the inner cylinder, 2, rolls such that the line of centers (in the x-y plane) makes angle θ (positive counterclockwise) to the vertical, then the initial point of contact o