10_P76InstructorSolution

10_P76InstructorSolution - 10.76 Model Model the sled as a...

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10.76. Model: Model the sled as a particle. Because there is no friction, the sum of the kinetic and gravitational potential energy is conserved during motion. Visualize: Place the origin of the coordinate system at the center of the hemisphere. Then 0 y R = and, from geometry, 1 y = cos . R φ Solve: The energy conservation equation 1 1 0 0 K U K U + = + is 2 2 2 1 1 0 0 1 1 1 1 1 cos 2 (1 cos ) 2 2 2 mv mgy mv mgy mv mgR mgR v gR φ φ + = + + = = (b) If the sled is on the hill, it is moving in a circle and the r -component of net F G has to point to the center with magnitude 2 net / . F mv R = Eventually the speed gets so large that there is not enough force to keep it in a circular trajectory, and that is the point where it flies off the hill. Consider the sled at angle . φ Establish an r -axis pointing toward the center of the circle, as we usually do for circular motion problems. Newton’s second law along this axis requires:
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