With Ch.8 problem 37, part (a), my understand is that, when the block is at the highest point on the track, it has some potential energy and zero kinetic energy. When it is at the bottom of the track, it has some kinetic energy and zero potential energy. When the block is at the verge of losing contact with the track at its highest point, my understanding is that the potential energy for when it is at the top of the track must equal the kinetic energy when it is at the bottom of the track. Thus, I get 2mgR=(1/2)mv^2. Solving for v, I get 2rootgR. However, I don't understand why the book has a factor of 5 under the square root. What's wrong?
The block loses contact with the track when the normal force applied by the track is just equal to zero. In this problem you use the law of conservation of energy to determine the magnitude of the velocity as a function of height (and vo), and then use Newton's second law to get an expression for the normal force.