9.66.Set Up:When it rolls without slipping,When there is no friction (on the smooth ice),and the rotational kinetic energy are constant. Use coordinates where is upward and at the bottom of thehill.Solve:Motion along the rough part: Let be the translational speed at the bottom of the rough part.so andMotion along the smooth part:gives9.67.Set Up:The two hands point in the same direction at midnight. They point in opposite directions whenwhere 1,Solve:The minute hand hasandand and the time is 12:33 a.m.and 38 min and the time is 1:38 a.m.and 44 min and the time is 2:44 a.m.and 49 min and the time is 3:49 a.m.and 54 min and the time is 4:54 a.m.and the time is 6:00 a.m.and 5 min and the time is 7:05 a.m.and 11 min and the time is 8:11 a.m.and 16 min and the time is 9:16 a.m.and 21 min and the time is 10:21 a.m.and 27 min and the time is 11:27 a.m.The same times repeat in the p.m.9.68.Set Up:The solid sphere has and the hollow spherical shell has Solve:so same Kand means and 9.69.
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