*44.  The drawing shows the top view of two doors. The doors are uniform and identical. Door A rotates about an axis through its left edge, and door B rotates about an axis through its center. The same force  is applied perpendicular to each door at its right edge, and the force remains perpendicular as the door turns. No other force affects the rotation of either door. Starting from rest, door A rotates through a certain angle in 3.00 s. How long does it take door B (also starting from rest) to rotate through the same angle?

 

w0441

 

 

 

Starting from rest and moving through an angle q in a time t, we use

 

 

So now consider the two doors for Door A we have

 

for Door B we have

 

Moment of inertia for Door A is the moment of inertia for a thin rectangular sheet on an end is

 

Moment of inertia for Door B is the moment of inertia for a thin rectangular sheet in the middle is

 

The Radius for the torques is for Door A the whole length of the Door L.  While for Door B the radius is half the door L/2.  So putting these into the angle formulas we get

 

And

 

Now the angle q is the same and we know F, L, and M are all the same so equating these two expressions

 

Simplifying

 

 

 

 

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This page last updated on January 11, 2020