Steel construction beams, with an industry designation of \"W12 22,\" have a wei

Question

Steel construction beams, with an industry designation of "W12 22," have a weight of 22 pounds per foot. A new business in town has hired you to place its sign on a 4.0 m long steel beam of this type. The design calls for the beam to extend outward horizontally from the front brick wall (see figure). It is to be held in place by a 5.0 m-long steel cable. The cable is attached to one end of the beam and to the wall above the point at which the beam is in contact with the wall. During the initial stage of construction, the beam is not to be bolted to the wall, but to be held in place solely by friction. (a) What is the minimum coefficient of friction between the beam and the wall for the beam to remain in static equilibrium? (b) What is the tension in the cable in this case? N

Explanation / Answer

Let the angle be Ø 22lbs/ft = 10kg/ft = 32.81kg/m The Co-efficient of friction µ = Frictional Force / Normal force Taking moments around the point on the wall find the tension in the cable Tc 32.81 x 9.8 x 4 x 2 = 4 x Tc sin Ø Tc = 643.1/sin Ø N The normal force on the wall Fn = Tc cos Ø = 643.1 cos Ø / sin Ø For the frictional force Ff take moments around the end of the beam 32.81 x 9.8 x 4 x 2 = Ff x 4 Ff = 643.1N so µ = Ff / Fn = 643.1 / 643.1 cos Ø / sin Ø µ = sin Ø / cos Ø = Tan Ø = 3/4 = 0.75

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