A uniform beam AB of mass 79.76 kg rests against the smooth vertical wall at A,

Question

A uniform beam AB of mass 79.76 kg rests against the smooth vertical wall at A, and the rough horizontal floor at B, as shown in the figure below. The coefficient of static friction between the beam and the horizontal floor is ? = 0.299.

Calculate the magnitude of the horizontal force P, in N, such that the end B has impending motion to the left. Give your answer to 3 decimal places. Assume g = 9.81 m/s2

Please show the working and answer for better understanding.

A uniform beam AB of mass 79.76 kg rests against the smooth vertical wall at A, and the rough horizontal floor at B, as shown in the figure below. The coefficient of static friction between the beam and the horizontal floor is ? = 0.299. Calculate the magnitude of the horizontal force P, in N, such that the end B has impending motion to the left. Give your answer to 3 decimal places. Assume g = 9.81 m/s^2

Explanation / Answer

is B is about to move to the left then friction points to the right and is at its max value
so we will have 5 forces:
P
normal at B, NB (points up)
friction at B, ?NB (points right)

weight, mg (points down)

normal at A, NA (points to the left)
so sum of forces in x:

P+?NB-NA=0

forces in y:

NB - mg =0 so NB = mg

sum of torques about B:

-mg 2.5 * cos(45) + NA * 5 * sin(45)=0

so NA*5*sin(45)=mg*2.5*cos(45)

NA = mg/2

combine these later two equations into the x equation

P + ?(mg) - mg/2 = 0

P = mg/2 - ?mg = (0.5 - 0.299)*79.76*9.81= 151.271

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