As shown in the figure below, a uniform beam is supported by a cable at one end

Question

As shown in the figure below, a uniform beam is supported by a cable at one end and the force of friction at the other end. The cable makes an angle of ? = 30°, the length of the beam is L = 3.75 m, the coefficient of static friction between the wall and the beam is ?s = 0.460, and the weight of the beam is represented by w. Determine the minimum distance x from point A at which an addional weight 2w (twice the weight of the rod) can be hung without causing the rod to slip at point A.

Ist time i submitted it the tutor got it wrong he put 2.1

Explanation / Answer

Horizontally, for tension T and normal force on beam Fn, we've got
Fn = Tcos30º = 0.866T

We know that at the threshold, Ff = µ*Fn = 0.460 * 0.866T = 0.3984T
where Ff is the friction force at the beam

Vertically, we've got Ff + Tsin30º = 0.3948T + 0.5T = 0.8948T = w + 2w = 3w
so T = 3.35w

Finally, consider the moment about the left end of the beam.
It must be zero, or the beam would rotate.
M = 0 = Tsin30º * L - w*L/2 - 2w*x = 3.35w * 0.5 * L - w*L/2 - 2w*x
3.35 * 0.5 * L -*L/2 - 2*x=0

4.40 = 2x
x = 2.2m

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