(a) To find the magnitude of the force in each of the struts AD and BD, resolve vertically:

\(500 + T \cos 45^{\circ} + T \cos 45^{\circ} - 100g = 0\)

Since \(T_A = T_B = T\), we have:

\(500 + 2T \cos 45^{\circ} = 100g\)

Solving for \(T\):

\(T \cos 45^{\circ} = \frac{100g - 500}{2}\)

\(T = \frac{500}{\sqrt{2}} = 354 \text{ N}\)

(b) To find the value of \(F\) for which the magnitude of the force in the strut AD is zero, resolve horizontally and vertically:

Vertically:

\(T_B \cos 45^{\circ} + 500 - 100g = 0\)

Horizontally:

\(F - T_B \sin 45^{\circ} = 0\)

Solving for \(F\):

\(F = T_B \sin 45^{\circ}\)

Since \(T_B \cos 45^{\circ} = 100g - 500\),

\(T_B = \frac{100g - 500}{\cos 45^{\circ}}\)

\(F = \frac{100g - 500}{\cos 45^{\circ}} \sin 45^{\circ} = 500 \text{ N}\)