Q $2.10$ points $-$ A uniform beam of length $L=10m$ and weight $W=500N$ is supported horizontally at one end by a hinge $($pivot$)$ and at the other end by a vertical cable. A $300$ N weight hangs from the beam at a distance of $6$ m from the hinge. Assume the beam is in static equilibrium.

$($a$)$ Using the First Condition for Equilibrium, write the equation representing the balance of forces acting on the beam.

$($b$)$ Using the Second Condition for Equilibrium, choose the hinge as the axis of rotation and write the torque equation to find the tension in the cable.

$($c$)$ Solve for the tension $T$ in the cable.

$($d$)$ Determine the vertical reaction force $F_{\text{h}\text{i}\text{n}\text{g}\text{e}}$ exerted by the hinge.

$($e$)$ Evaluate the stability of the system. Would the system remain stable if the weight were moved to $8$ m from the hinge? Assume the cable can withstand up to $500$ N in tension. Justify your answer qualitatively.