The drawing shows a uniform horizontal beam attached to a vertical wall by a fri

Question

The drawing shows a uniform horizontal beam attached to a vertical wall by a frictionless hinge and supported from below at an angle = 43o by a brace that is attached to a pin. The beam has a weight of 340 N. Three additional forces keep the beam in equilibrium. The brace applies a force Upper P Overscript bar EndScripts to the right end of the beam that is directed upward at the angle with respect to the horizontal. The hinge applies a force to the left end of the beam that has a horizontal component Upper H Overscript bar EndScripts and a vertical component Upper V Overscript bar EndScripts. Find the magnitudes of these three forces.

Explanation / Answer

Clockwise : mg x ½L = Anti-clock : Pv x L .. .. Pv = vert. component of P
Pv = ½ mg .. .. ½ x 340N .. .. Pv = 170 N

• Vertical forces are in balance .. 340N = Pv + V
V = 340 - Pv .. .. (340 - 170)N .. .. .. V = 170 N

sin(43) = Pv/P
P = Pv/sin(43) .. .. 170N/sin(43) .. .. .. P = 249.26 N

• Use Pythagorean Theorem to solve for H
P^2 = sqrt(249.26^2 + H^2) ........ H = sqrt(P^2 - V^2) ...... H= 182.29 N

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