Problem 1: A pendulum is formed from a small ball of mass m on a string of length L As Fig. 1 shows. a peg is height h : L/3 above the pendulum's lowest point. From what minimum angle 8 must the pendulum be released in order for the ball to go over the top of the peg without the string going slack? a) The minimum speed at which the ball can pass over the top of the peg without the string going slack corresponds to the case when the gravitational force alone is barely suicienl to provide the required centripetal acceleration a, at that point (for larger speeds, the tension in the striIn/gswill also contribute to a,). Show that this minimum speed FIG. 1: The scheme for Problem 1 IS I} = g . b) The energy conservation requires K; + U1 = Kf + Uf, where the initial state is when the pendulum is at angle (9 with respect to the vertical (K, = 0), and the nal state is when the hall is passing over the top of the peg. Using the expression for u from the previous step to compute Kf> show that the string will not go slack if9 is larger than arccos % = 80A\". Problem 2: An m = 10 kg box slides 40 m down the frictionless ramp Show in Fig. 2, then collides with a spring whose spring constant is k = 250 N/ m. At what compression of the spring does the box have its maximum speed? a) Fig, 3 shows the energy diagram of the system. The origin of y-axis is chosen to be at the point where the box rst contacts the spring (see the yaxis in Fig. 2). For y > 0, the potential 411 m energy is entirely gravitational. For y 0 myyMSYzMWH) - fory