A smooth rod of length l is kept inside a trolley at an angle θ as shown in the figure. What should be the acceleration a of the trolley so that the rod remains in equilibrium with respect to it?

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Forces on the rod (in the trolley frame, add the pseudo force $ma$ backward at the centre): wall reaction $N_{1} = ma$ at the upper end (horizontal), floor reaction $N_{2} = m g$ at the lower end.

Torque about the lower end of the rod, of length $L$ : $m g \frac{L}{2} cos θ + ma \frac{L}{2} sin θ - N_{1} L sin θ = 0$ .

Using $N_{1} = ma$ : $m g \frac{L}{2} cos θ = ma \frac{L}{2} sin θ \Rightarrow a = g cot θ$ .

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