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 0yR=and, from geometry, 1y=cos .RφSolve:The energy conservation equation 11 0 0KUKU+=+is 2221100111cos2(1 cos )2mvmgymvmgymvmgRmgRvgRφφ+=+⇒+=⇒=−(b)If the sled is on the hill, it is moving in a circle and the r-component of netFGhas to point to the center with magnitude 2net/.FmvR=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 .Establish an r-axis pointing toward the center of the circle, as we usually do for circular motion problems. Newton’s second law
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