As depicted in the figure, a rough track features a portion shaped like a cylinder of radius R . Find the minimum linear speed at which a sphere of radius r should be set rolling on the horizontal segment so that it completely traverses the circle on the cylindrical part.

Step-by-step solution

For the sphere to completely traverse the circular track, it must not lose contact at the highest point. At the highest point, the minimum required speed v_ top is obtained when the normal force is zero. The equation of motion for the center of mass at the highest point is: mg = mv_ top ^2 R - r v_ top ^2 = g(R - r) The kinetic energy of a rolling solid sphere is: K = 1 2 mv^2 + 1 2 I ^2 = 1 2 mv^2 + 1 2 ( 2 5 mr^2 ) ( v r )^2 = 7 10 mv^2 The height of the center of the sphere at the highest point relative to its i