Rotating Door
المؤلف:
Sidney B. Cahn Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 1 , p 22
1-8-2016
1301
Rotating Door
A uniform rectangular door of mass m with sides a and b (b > a) and negligible thickness rotates with constant angular velocity ω about a diagonal (see Figure 1.1). Ignore gravity. Show that the torque |N| = [m(b2 – a2)abω2]/[12(a2 + b2)] must be applied to keep the axis of rotation fixed.
Figure 1.1
SOLUTION
We will use the frame rotating with the door (body frame, axes 
see Figure 1.2), so that we can use the Euler equations
(1)
(2)
(3)
Figure 1.2
Where ω1, ω2, ω3 are the components of angular velocity in this frame and I1, I2, I3 are the principal moments of inertia. In this case I1 and I2 correspond to the moments of inertia of rods of length a and b respectively, which can be calculated easily:
Since the problem is two-dimensional (we can disregard the thickness of the door),
In our frame ω3 = 0 and ω1 and ω2 are constant.
(4)
(5)
Substituting (4) and (5) into (1)–(3) we obtain
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