Discussion
One end of a uniform rigid rod \(OA\) of length \(L\) and mass \(m\) is attached to a frictionless hinge at \(O\). The other end of the rod is connected to the roof at \(B\) with a mass-less inextensible thread \(AB\). Initially the rod is horizontal and at rest. The gravity is acting vertically downward as shown. Immediately after the thread \(AB\) is cut, the reaction on the rod at \(O\) is
Answer: A
The reaction is \(\frac{mg}{4}\) in the positive y-direction.
Just after the cut the rod is still at rest, so only the angular acceleration about O matters. Taking moments about the hinge:
\[ \begin{aligned} \frac{mL^{2}}{3}\alpha &= mg\frac{L}{2},\quad \alpha = \frac{3g}{2L} \\ a_{G} &= \alpha\frac{L}{2} = \frac{3g}{4}\ \text{(downward)} \end{aligned} \]
Newton's law for the vertical motion of the centre of mass gives \(mg - R = m\cdot 3g/4\), so \(R = mg/4\), upward. mg would be the static reaction with no thread, and the downward options have the wrong sense: the hinge must still support part of the weight.
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