6) To form a pendulum, a 0.092 kg ball is attached to one end of a rod of length 0.62 m and negligible mass, and the other end of the rod is mounted on a pivot. The rod is rotated until it is straight up, and then it is released from rest so that it swings down around the pivot. When the ball reaches its lowest point, what are (a) its speed and (b) the tension in the rod? Next, the rod is rotated until it is horizontal, and then it is again released from rest. (c) At what angle from the vertical does the tension in the rod equal the weight of the ball? (d) If the mass of the ball is increased, does the answer to (c) increase, decrease, or remain the same?

13.39 • CALC Consider the ring- shaped object in Fig. E13.39 D. A particle with mass m is placed a distance x from the center of the ring, along the line through the center of the ring and perpendicular to its plane. (a) Calculate the gravitational potential energy U of this system. Take the potential energy to be zero when the two objects are far apart. (b) Show that your answer to part (a) reduces to the expected result when x is much larger than the radius a of the ring. (c) Use F. = -dU/dx to find the magnitude and direction of the force on the particle (see Section 7.4 9). (d) Show that your answer to part (c) reduces to the expected result when x is much larger than a. (e) What are the values of U and F, when x = 0? Explain why these results make sense. Figure E13.39 т M

The orienting track of a bowl feeder consists of one device (X). The part has four natural resting positions. The device efficiency matrix and the initial distribution matrix are below. Parts enter the track at the rate of 170 min. How many parts exit device-X in position 3 per minute? (One place of decimal) INITIAL DISTRIBUTION MATRIX Position P1 P2 P3 P4 Bowl 0.41 0.18 0.29 0.120 DEVICE-X TRANSITION MATRIX 0.50 P13 0.49 P23 0.19 0.29 P33 0.18 0.37 P43 0.15 1 Note: Each row must sum to 1. Therefore P13, P23, P33 AND P43 ARE 1-Sum(All other row cells)

A spring with spring constant k and equilibrium length zero is attached tothe top of a frictionless hoop of radius R. The spring is stretched and connected to a bead of mass m at the bottom of the hoop. At t = 0 the bead is given an initial speed v0 and the bead moves up the hoop. Find the speed of the bead as a function of position on hoop v(θ).