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2. The purpose of this problem is to compare overhanging simple support and cantilever support, with the ultimate conclusion that a roller support right next to a pinned support is effectively the same as a cantilever support. (a) (3 points) Consider a simply-supported overhanging beam of length L, with a pinned support at x = 0 and a roller support at x = Ax (where 0 < Ax < L, so the overhang is L - Ax), subject to a spatially-uniform transverse load p(x) = P > 0. Sketch this beam and solve for V(x) and M(x). (b) (1 point) Now do the same for another beam with the same external transverse loading, but cantilever support at x = 0 and a free end at x = L. (Recall: A very similar example was already worked in lecture, with P = -PgA.) (c) We now consider the limit of Ar → 0 in the beam from Problem 2a, and show that it behaves similarly to a cantilever beam for small Ax. i. (2 points) On one set of axes, sketch M(x) for the cantilever beam (using a dashed line), then M(x) for the simply-supported beam with Ar = L/2, L/4, and L/8 (using solid lines). The results for the simply-supported beam should visually approach those of the cantilever beam as Ax → 0. ii. (1 point) Evaluate the moment produced by the force couple consisting of the pinned and roller sup- ports in the simply-supported beam. Hint: This moment should match the reaction of the cantilever support.