From
Solution 557, tension governs at both positive and negative maximum moments.
At M = -0.5wo x2 N·m:
20=0.5wox2(180)(1000)36×106
wo=8000/x2
At M = -0.5wo x2 + 0.5wo(5 - x)2 N·m:
20=[0.5wox2+0.5wo(5−x)2](50)(1000)36×106
14400=−0.5wox2+0.5wo(5−x)2
28800=−wox2+wo(5−x)2
28800=−wox2+wo(25−10x+x2)
28800=−wox2+(25−10x)wo+wox2
28800=(25−10x)wo
28800=(25−10x)(8000/x2)
(28800x2/8000)=25−10x
185x2=25−10x
18x2=125−50x
18x2+50x−125=0
x=1.59 m and −4.37(absurd)
use x = 1.59 m answer
wo=8000/1.592
wo=3164.43N/m
wo=3.16kN/m answer