15 a top spinning on the floor precesses because the

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15. A top spinning on the floor precesses because the torque due to gravity, about the point of contact of the top with the floor, causes the angular momentum of the top to change. The direction of this torque is: (a) parallel to the angular momentum. (b) parallel to the angular velocity vector. (c) parallel to the axis of rotation. (d) perpendicular to the floor. (e) parallel to the floor. Answer is E since: Since τ = r × F , torque acts in a direction perpendicular to both the radius and the force. The force in this case is gravity, which points down, while the radius points along the axis of rotation. Using the right hand rule gives a direction parallel to the floor.
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16. A hoop, a uniform disk, and a uniform sphere, all with the same mass and outer radius, start on a horizontal plane with the same translational motion speed of their respective centers of mass and roll without sliding. Then, they roll up an identical incline. Rank the objects according to the maximal height they can reach, from the lowest to the highest height. (a) hoop, disk, sphere (b) disk, hoop, sphere (c) sphere, hoop, disk (d) sphere, disk, hoop (e) hoop, sphere, disk Answer is D since: When the objects stop, all their kinetic energy has been converted to potential, meaning K = mgh . Therefore, whichever has the highest kinetic energy will go the highest. K = 1 2 mv 2 + 1 2 2 = 1 2 mv 2 + 1 2 I ( v/r ) 2 . Since each has the same velocity and radius, whichever object has the largest I also has the largest K . I hoop = MR 2 > I disk = 1 2 MR 2 > I sphere = 2 5 MR 2 . Answer choice D has the objects ordered from lowest to highest I . 7

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