Homework #7
Assigned: Feb. 27th, 2007
Due: Mar. 6th, 2007
Points: 20

1. 10 points
   A boat needs to cross a river of width, L. Assume that the speed of the boat is v0. Assume also the water in the river has a velocity profile, w(x). Here, x is the shortest line joining the two shores of the river and hence is perpendicular to the shores. Formulate the problem to minimize the time to cross the boat and to find the path that the boat should take. The steering angle alpha(x) may be taken as the variable in optimization. Write the Euler-Lagrange equations and solve them to the extent possible. Sketch a possible solution to this problem.
2. 10 points
   A beam of uniform cross-section with elasticity modulus, EI, and length, L, is fixed at one end. The other end is guided on a circulat arc whose center lies on the extended axis of the beam at a distance of (L+R) from the fixed end. There is point transverse force P at the mid-point of the beam. There is also a point force along the axis of the beam located at the guided end and directed away from the fixed end. Use the minimum potential energy principle and write down the equilibrium equation and the boundary conditions.