1. A ball attached to the end of a string is swung in a vertical circle. Assuming the energy of the ball-Earth system remains constant, show that the tension in the string at the bottom is larger than the tension in the string at the top by six times the weight of the ball. 2. A boy is seated at the top of a block of ice in the shape of a hemisphere. Model the boy as a particle and the ice as frictionless. As the boy slides down the side of the hemisphere, determine the height at which he loses contact with the ice. Express as a function of R. 3. A rope of length L, mass M, and constant mass density rests coiled on a frictionless tabletop. Determine the work required to lift the entire length of string off the table at a constant speed. Express in terms of M, g, and L. 4. (Optional) A bowling ball of mass m is dropped from rest above the opening of a tube. It falls a distance d before encountering the tube, though which it slides frictionlessly. The semicircular tube is fastened to a support by two bolts, each of which can sustain a maximum force of F. What is the maximum distance, d, from which the ball can be released above the opening of the tube so the bolts do not fail? Express d as a function of m, g, r, and F. 5. (Optional) A small mass is attached to two identical springs, of force constant k, on a fric- tionless, horizontal tabletop. When the mass is at the origin, both springs are at their natural lengths. The mass is pulled a distance r in the direction perpendicular to the original orienta- tion of the springs (see figure). (a) Show that the net force on the mass from the springs is F = -2kx (1- L VI2 + 125. (Optional) A small mass is attached to two identical springs, of force constant k, on a fric- tionless, horizontal tabletop. When the mass is at the origin, both springs are at their natural lengths. The mass is pulled a distance r in the direction perpendicular to the original orienta- tion of the springs (see figure). (a) Show that the net force on the mass from the springs is F = -2kx (1 - VF +1 (b) Show that the potential energy of the system is given by U(x) = kx2 + 2kL(L - Vx2+ 12) where the potential at the origin is zero. (c) If the mass is released from rest at a distance of 0.50 m from the origin, what is the speed of the mass as it passes through the origin? Use the following values: k = 40.0 N/mm, L = 1.20 m, and m = 1.0 kg. Overhead view