Opening a Door

In the \(xyz\)-coordinate system, a \(20 \text{ kg}\) door is hinged so that it can rotate freely about the \(z\)-axis (its length is along the \(z\)-axis). The door is \(1 \text{ m}\) wide, and its width is initially aligned with the \(x\)-axis. The door is initially at rest.

A constant force \((F_x,F_y) = \big(-10 \text{ N}, +10 \sqrt{3} \text{ N}\big) \) is applied to the door at its far edge (farthest from the \(z\)-axis).

What is the door's angular speed \((\)in \(\text{rad/s})\) when its width is first aligned with the \(y\)-axis?

Details and Assumptions:

  • Neglect gravity and air resistance.
  • The door's mass is uniformly distributed over its width.
  • Give your answer to 3 decimal places.

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