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