One end of a uniform 350mlong rod of weight Fg is supported

One end of a uniform 3.50-m-long rod of weight F_g is supported by a cable at an angle of theta = 37 degree 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 mu_s = 0.505. 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___m.

Solution

F_x : N T cos theta = 0

F_y : T sin theta + F_R 2 * w = 0

: T * L sin theta w * L / 2 w * x = 0

where torque has been calculated around point A. From the first two equations we have

2 * w = T sin theta + F_R < T sin theta + mu_s * T cos theta

T sin theta > 2 * w / 1 + mu_s * cot theta

from the third equation we find

T sin theta = w * (1/2 + x/L)

Putting these together we have

x > L [( 2 / 1 + mu_s * cot theta) - 1/2]

= 3.5 [( 2 / 1 + 0.505* cot 37) - 1/2]

= 2.44 m