One end of a uniform 390mlong rod of weight Fg is supported

One end of a uniform 3.90-m-long rod of weight F_g is supported by a cable at an angle of = 37° with the rod. The other end rests against the wall, where it is held by friction as shown in the figure below. The coefficient of static friction between the wall and the rod is _s = 0.540. Determine the minimum distance x from point A at which an additional object, also with the same weight F_g, can be hung without causing the rod to slip at point A.

Solution

x = xmin , the rod is on the verge of slipping

f = fsmax = mu_s*N = 0.540N

in x direction sum of all force = 0

N - Tcos37 = 0

N = 0.799T

f = 0.43146T

in y direction

Fy = 0

f + Tsin37 - 2Fg = 0

0.43146T + 0.602T -2Fg = 0

T = 1.935Fg

torque = 0 for an axis perpendicular to the page and through the left end of the beam gives

-Fg*xmin - Fg*1.95 + [ 1.935*Fg sin37 * 3.9] = 0

xmin = 2.59 m