A 2.0m-long uniform beam is hinged to a vertical wall at its lower end (Point O). The beam makes an adjustable angle θ with the horizontal, where 0∘≤θ≤90∘. A worker attempts to support the beam by tying a light rope from a point 1.5m up the beam to a hook directly above Point O. The rope is always kept taut and forms an angle ϕ with the beam at the attachment point.
Assume the beam’s weight of 120N acts at its midpoint.
a. Part 1: Derive an expression, in terms of θ, for the torque about Point O produced by the beam’s weight.
b. Part 2: Show that there exists an orientation θ for which the torque from the rope about Point O is exactly zero, regardless of the rope’s tension. Provide a clear explanation without using the explicit torque formula.
c. Part 3: When θ=40∘, determine the angle ϕ that the rope must make with the beam so that the rope’s pull contributes no torque about Point O.
d. Part 4: Explain qualitatively how your result in part (c) would change (if at all) if the rope were attached at the beam’s midpoint instead of 1.5m from Point O.