A cart of mass Mis on a level, horizontal track. The cart is attached to one end of an ideal spring, and the other end of the spring is attached to a vertical support. The spring has a spring constant k. The cart is pulled to the right, stretching the spring a distance d, as shown above, and released from rest at time t = 0. The cart-spring system begins to oscillate, and the position x of the cart as a function of time t is given by the equation x(t) = d cos(wt) , where x is in meters, w is in rad/s, and tis in seconds. Assume the mass of the cart's wheels to be negligible. (a) Derive an equation for the velocity of the cart vas a function of time t. (b) On the axes below, sketch a graph of the velocity v of the cart as a function of time t for two complete oscillations. On the graph, T represents the time for one complete oscillation. Explicitly label asymptotes, maxima, or minima with algebraic expressions, as appropriate. -+- 2T(c) Express all answers in part (c) in terms of w, t, M, d, k, and physical constants, as appropriate. i. Determine an expression for the kinetic energy K of the system as a function of time t. ii. Determine an expression for the potential energy U of the system as a function of time t. iii. Using the equations from parts (c)i and (c)ii, show that the rate of change of the total energy of the system dE/dt is zero. M Figure 1 1-d-- Clay M Figure 2Two springs identical to the spring above are attached to each side of the cart of unknown mass M, as shown in Figure 1 above. The cart is on a level, horizontal track of negligible friction. A piece of clay is added to the top of the cart so that it will stick to the cart. The cart is displaced to the left a distance dand released from rest, as shown in Figure 2 above. The period of oscillation is recorded. This procedure is repeated for several different pieces of clay of different masses. In each trial, the cart is displaced the same distance d. The data are shown below on the linear graph of T2as a function of Inc, where Tis the period of oscillation and mo represents the mass of the clay. - a (d) From the graph, determine the following. i. k, the spring constant of each spring ii. M, the mass of the cart 9 ' (e) The experiment is repeated, but in a second set of trials, the cart is pulled back a distance D, where D > d. Describe any changes that will occur for the straight line on the graph in part (d). Justify your