Find the length of the third segment in tems of L?

Question

A thin uniform metal rod is bent into three perpendicular segments, two of which have length L. You want to determine what the length of the third segment should be so that the unit will hang with two segments horizontal when it is supported by a hook

http://session.masteringphysics.com/problemAsset/1039098/5/YF-11-36.jpg

Find x in terms of L.

Answer 1

As the wire is uniform, the length of a segment is proportional to its mass.

Moments about the pivot for the three segments are as follows.

Upper segment: L * L / 2 clockwise,

Vertical segment: L * L clockwise,

Lower segment: x(x / 2 - L) anticlockwise.

Equating the total of these moments to zero:

3L^2 / 2 = x(x / 2 - L)

3L^2 = x^2 - 2xL

x^2 - 2Lx - 3L^2 = 0

x = [ 2L +/- sqrt(4L^2 + 12L^2) ] / 2

= L +/- 2L

The positive root is x = 3L.