Consider the spring-mass system, shown in Figure 4.2.4, consisting of two unit masses suspended from springs with spring constants 3 and 2, respectively. Assume that there is no damping in the system. a. Show that the displacements u1 and u2 of the masses from their respective equilibrium positions satisfy the equations u__ 1 + 5u1 = 2u2, u__ 2 + 2u2 = 2u1. (22) b. Solve the first of equations (22) for u2 and substitute into the second equation, thereby obtaining the following fourth-order equation for u1: u(4) 1 + 7u__ 1 + 6u1 = 0. (23) Find the general solution of equation (23). c. Suppose that the initial conditions are u1(0) = 1, u_ 1(0) = 0, u2(0) = 2, u_ 2(0) = 0. (24) Use the first of equations (22) and the initial conditions (24) to obtain values for u__ 1(0) and u___ 1 (0). Then show that the solution of equation (23) that satisfies the four initial conditions on u1 is u1(t) = cos t. Show that the corresponding solution u2 is u2(t) = 2 cos t. d. Now suppose that the initial conditions are u1(0) = 2, u_ 1(0) = 0, u2(0) = 1, u_ 2(0) = 0. (25) Proceed as in part c to show that the corresponding solutions are u1(t) = 2 cos 6 t_ and u2(t) = cos 6 t_. e. Observe that the solutions obtained in parts c and d describe two distinct modes of vibration. In the first, the frequency of the motion is 1, and the two masses move in phase, both moving up or down together; the second mass moves twice as far as the first. The second motion has frequency6, and the masses move out of phase with each other, one moving down while the other is moving up, and vice versa. In this mode the first mass moves twice as far as the second. For other initial conditions, not proportional to either of equations (24) or (25), the motion of the masses is a combination of these two modes.

Lecture 6: Limits, Velocity and Tangent Lines (Sections 2.1 and 2.2) Recall that the average velocity = Note that velocity has direction and speed = |velocit|. Now try the following example before class: ex. At limigngaphottitce from a ﬁxed starting point is given...