The uniform garage door has a mass of 150 kg and is guided along smooth tracks at its

Chapter 18, Problem 18-51

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The uniform garage door has a mass of 150 kg and is guided along smooth tracks at its ends. Lifting is done using the two springs, each of which is attached to the anchor bracket at A and to the counterbalance shaft at B and C. As the door is raised, the springs begin to unwind from the shaft, thereby assisting the lift. If each spring provides a torsional moment of \(M = (0.7 \theta) \ N \cdot m\), where \(\theta\) is in radians, determine the angle \(\theta_{0}\) at which both the left-wound and right-wound spring should be attached so that the door is completely balanced by the springs, i.e., when the door is in the vertical position and is given a slight force upward, the springs will lift the door along the side tracks to the horizontal plane with no final angular velocity. Note: The elastic potential energy of a torsional spring is \(V_{e}=\frac{1}{2} k \theta^{2}\), where \(M=k \theta\) and in this case \(k=0.7 \ \mathrm{N} \cdot \mathrm{m} / \mathrm{rad}\).