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Analyzing Rigid Body Equilibrium Using Trigonometry: A Step-by-Step Guide

Understand the static equilibrium of a system with multiple forces using trigonometric calculations. Calculate unknown forces and angles based on known parameters such as weight, dimensions, and angles. Solve for pivot forces and angles to ensure balanced equilibrium in a simple and concise manner.

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Analyzing Rigid Body Equilibrium Using Trigonometry: A Step-by-Step Guide

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  1. Rigid Body Equilibrium By Leo Takahashi, PSU Beaver Campus

  2. PB = L ; PA = D ; PC = S Weight of Sign = W B Weight of Pole = mg A steel cable  Thin chain Pole’s Center of gravity  C P Frictionless Pivot

  3. PB = L ;PA= D ; PC = S B Weight of Sign = W Weight of Pole = mg A   +x B C P FT A  W  +y C fx fy mg  P

  4. PB = L ; PA = D ; PC = S +x B Fx = 0 Fy = 0 FT A  W +y C P = 0  fx fy mg C = 0 A = 0 B = 0  P

  5. +x L FT  W D +y S fx fy mg   Fx = 0 P fx – mgCos - FTCos - WCos = 0 Fy = 0 fy – mgSin + FTSin - WSin = 0 P = 0 -(mgS)Sin + FTDSin - WLSin = 0

  6. If m, g, W, S, D, L, , and  are known, this set of equations can be solved for the three unknowns, fx , fy, and FT. Fx = 0 fx – mgCos - FTCos - WCos = 0 The magnitude of the pivot force can be calculated from its components, and the angle the pivot force makes with the pole (+x axis) is P= Tan-1(fy/fx). Fy = 0 fy – mgSin + FTSin - WSin = 0 P = 0 -(mgS)Sin + FTDSin - WLSin = 0

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