Question about linear fictitious force, (link in the description)?

In your equation, I suppose the parameter θ is the angle between the rod and the top of the car. In so doing, your equation is actually describing an oscillation about the top of the car (because θ is measured from the top of the car).

But the rod is, in fact, oscillating about its equilibrium position, which is at an angle θ with the top of the car.

Moreover, according to your equation, the force mg.sin(θ) and the force mA.cos(θ) are acting along the length of the rod, but in opposite directions. Since the two forces pass through the pivot, there should be no torque produced. But you have made a wrong assumption that the two forces are acting perpendicularly to the rod.

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