Example #8: A block of mass M is released from rest at point 1, as shown in the figure. The block slides without frictional forces along the circular arc but encounters frictional forces as soon as it reaches the horizontal portion of the track at point 2. The block travels a distance D along the horizontal track before coming to rest at point 3. R D Consider the block-Earth system. In terms of the mechanical energy of the system, which of the following claims is correct, and why? (Click HERE to verify your answer). a. The system is open, because there is a net force exerted on the block. b. The system is open, because the block's velocity is zero at points 1 and 3. c. The system is closed, because there is a net force exerted on the block. d. The system is closed, because the block's velocity is zero at points 1 and 3.Example #9: Objects X and Y are connected by a string of negligible mass and suspended vertically over a pulley of negligible mass, creating an Atwood's machine, as shown in the figure below. The objects are initially at rest, and the mass of Object Y is greater than the mass of Object X. As Object Y falls, how does the gravitational potential energy of the Object X-Object Y-Earth system change? All frictional forces are considered to be negligible. (Click HERE to verify your answer). a. The gravitational potential energy increases because the center of mass of Object X and Object Y moves upward. b. The gravitational potential energy increases because the center of mass of Object X and Object Y moves downward. Object Y c. The gravitational potential energy decreases because the center of mass of Object X and Object Y moves upward. d. The gravitational potential energy decreases because the center of mass of Object X and Object Y moves downward. Object X 18Example #10: A block of mass m is launched by a spring of negligible mass along a horizontal surface of negligible friction. The spring constant of the spring is k. The spring is initially compressed a distance x.. The block is released from rest. Some time after the block is released and travels in the direction shown in the figure, the spring compression is x, . Unstretched Spring XO DOOD V= 0 Xf Which of the following mathematical calculations can a student use to determine the speed v, of the block at this new position? (Click HERE to verify your answer). a k(x. ) _ m (v / 2 ) 2 2 b. k(xo ) _ k(x2) m(v2) 2 2 2 C. F(x) = _m(4vz) 2 , where F = k(x.) d. F(x)cose = m(4u?) 2 - , where F = k(x.)Example #11: A student conducts an experiment in which an object is released from rest and falls to the floor. In the experiment, frictional forces CANNOT be neglected. The student uses experimental data to create two graphs. Figure 1 is a graph of kinetic energy of the object as a function of time. Figure 2 is a graph of the object-Earth system's gravitational potential energy as a function of time. 0.7 0.6 0.5- 0.4 Gravitational Potential Energy (J) Kinetic Energy (J) 0.3 0.1- 2 6 2 6 Time (s) Time (s) Figure 1 Figure 2 How should the student use one or both graphs to determine how much the total mechanical energy (ATME) changes after 5.0 s? (Click HERE to verify your answer). a. Calculate the magnitude of the difference between the final kinetic energy of the object and the final kinetic energy of the object by using the graph in Figure 1. b. Calculate the magnitude of the difference between the final gravitational potential energy of the system and the initial gravitational potential energy of the system by using the graph in Figure 2. c. Calculate the magnitude of the difference between the final kinetic energy of the object found from the graph in Figure 1 and the final gravitational potential energy of the object- Earth system found from the graph in Figure 2. d. Calculate the magnitude of the difference between the final kinetic energy of the object found from the graph in Figure 1 and the initial gravitational potential energy of the object- Earth system found from the graph in Figure 2