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.