To show that the forces are in equilibrium, we need to resolve the forces horizontally and vertically and show that the net force in each direction is zero.
Resolving horizontally:
\(X = 78 \times \frac{5}{13} - 50 \times \frac{3}{5} = 78 \cos 67.4^\circ - 50 \cos 53.1^\circ\)
\(X = 30 - 30 = 0\)
Resolving vertically:
\(Y = 78 \times \frac{12}{13} + 50 \times \frac{4}{5} - 112 = 78 \sin 67.4^\circ + 50 \sin 53.1^\circ - 112\)
\(Y = 72 + 40 - 112 = 0\)
Since both \(X = 0\) and \(Y = 0\), the forces are in equilibrium.
Alternatively, using Lami's theorem:
\(\frac{112}{\sin 59.5^\circ} = \frac{50}{\sin 157.4^\circ} = \frac{78}{\sin 143.1^\circ}\)
Calculating the exact values:
\(\frac{112}{56/65} = \frac{50}{5/13} = \frac{78}{3/5} = 130\)
Thus, the forces satisfy Lami's theorem, confirming equilibrium.
