Relabel the figure as:

From the equilibrium condition:

$\overline{){\mathbf{\Sigma}}{\mathbf{F}}{\mathbf{=}}{\mathbf{0}}}$

The vertical equilibrium:

The varying load given is divided into a rectangle and a triangular load.

We have:

w_{1}L + (1/2)(w_{2} - w_{1})L = P + 2P

w_{1}L + (1/2)w_{2}L - (1/2)w_{1}L = 3P

(1/2)w_{1} + (1/2)w_{2}L = 3P/L

w_{1} + w_{2} = 6P/L

Considering the moment at B:

w_{1}L + (1/2)L + (1/2)(w_{2}_{ }- w_{1})L + (1/3)L = P(2/3)L + 2P(1/3)L

w_{1}L^{2}/2 + w_{2}L^{2}/6 - w_{1}L^{2}/6 = 4PL/3

2w_{1} + w_{2} = 8P/L

If P = 500 lb and L = 12 ft., what are the magnitudes of w_{1}, w_{2} ?

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