What's New X Assignment 4 X Course Hero x | + X Problem 2. Consider an elastic membrane D fixed at its boundary bud(D). The potential energy of the stretched membrane is defined through the relation dU xdS - dA, where dU is the differential of the potential energy, dS is the surface differential in the stretched state, and dA is the area differential in the resting state. Recall that a quantity A is proportional to another quantity B, denoted as Ac B, if A=kB. Assume that the stretch function is given by z = u(x, y). a) Show that for sufficiently small u, the following relation holds du x-Vu|2 dA. Hint: you may wish to use the linear approximation V Ity = 1+zy , when y is small. b) The total potential energy of the membrane can be calculated as U = du . Let D be a membrane in the shape of the unit disk stretched according to the function u(x, y) = sin(7(1 -22 - y2)). Find the total potential energy of the membrane if dU = 2 | Vu|2 dA for a positive constant k. Q Search ENG 8:21 PM US 2023-11-02