6 Minute Solutions, MSM Problem 39

This problem essentially breaks down to a beam bending problem. The cantilever beam is 2 inches long with a force applied at 1.5'' from the supported end. The problem states that the beam must not deflect more than a given amount at the point where the force is applied.

The following equation is used in the solution to calculate the deflection at the point of the application of the force:



However, based on my understanding this equation should be used for the deflection at the tip of the beam. For the deflection at point x, this equation should (I believe) be used:



Am I off base here or is the solution in the book incorrect? The book found the required diameter of the beam to be 0.442 inch while my solution was 0.3446 in. Since the question asked to find the smallest standard diameter, both these round up to 0.50 in so I was able to obtain the correct answer, but I would still like some validation as to whether or not my solution approach is correct.

The first equation is valid for the force being placed at the end of the beam.  The correct equation for force placed some distance a from the supported end is:

y max=[(Fa^2)/(6EI)]*[a-3L]  where L is the total beam length.

The equation used in 6MS is also valid for the "deflection - at point of load" for the cantilever, maybe not call it "y max" but "y at point of load". y at point of load = Pb^3/(3EI) and b is 1.5 in