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Equilibrium

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The weight $W=\\var{W}$N is supported by the rope-and-pulley arrangement shown. Knowing that $\\beta=\\var{b}^\\circ$, determine the magnitude (in N) and direction (angle $\\alpha$ in degrees) of the force $\\mathbf{P}$ that must be exerted on the free end of the rope to maintain equilibrium.

Free-Body Diagram: Pulley A

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$\\overset{+}{\\rightarrow}\\sum F_x=0$: $\\quad$  $2P \\sin \\beta - P \\cos \\alpha=0$

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$\\cos \\alpha=2\\sin\\beta=2\\sin \\var{b}^\\circ, \\quad\\Rightarrow\\quad \\alpha=\\var{a}^\\circ\\quad\\blacktriangleleft$

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$+\\!\\!\\uparrow\\sum F_y=0$: $\\quad$  $2P\\cos \\beta+P\\sin\\alpha - W=0$

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$2P\\cos \\var{b}^\\circ+P\\sin\\var{a}^\\circ=\\var{W}\\,\\text{N}$

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$P=\\dfrac{\\var{W}\\,\\text{N}}{2\\cos \\var{b}^\\circ+\\sin\\var{a}^\\circ}=\\var{P}\\,\\text{N}\\quad\\blacktriangleleft$

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direction (angle $\\alpha$):

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magnitude:

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