Question

A wooden bridge crossing a canyon consists of a plank with length density λ = 2.25 kg/m suspended at h = 10.89 m below a tree branch by two ropes of length L = 2h and with a maximum rated tension of 2000 N, which are attached to the ends of the plank, as shown in the figure. A hiker steps onto the bridge from the left side, causing the bridge to tip to an angle of 22.3° with respect to the horizontal. What is the mass of the hiker?

Answer 1

If the two lines are attached at the same point on the branch so a triangle is formed, then the center of gravity of the plank, hiker system must be directly below the attachment point.

mass of plank = 2.25 * length
for length (.5L)^2 = 21.78^2 - 10.89^2
(.5 L)^2 = 474.4 - 118.6
.5 L = 18.9
L = 37.7 meters long
so mass of plank = 2.25*37.7 = 84.9 kg

now balance
how far is G from center of plank?
tan(22.3) = x/10.89
x = 4.47
so how far is left end from G?
18.9 - 4.47 = 14.4

so m(14.4) = 84.9(4.47)
m = 26.4 kg