A car door is inadvertently left slightly open when the brakes are applied to give the car a constant rearward acceleration \(a\). Derive expressions for the angular velocity of the door as it swings past the \(90^{\circ}\) position and the components of the hinge reactions for any value of \(\theta\). The mass of the door is \(m\), its mass center is a distance \(\bar{r}\) from the hinge axis \(O\), and the radius of gyration about \(O\) is \(k_{O}\).

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