A truck at rest has one door fully open as shown in figure
a)
1 The truck accelerates forward at a constant rate A, and the door begins to swing
shut. The door is uniform and solid, with height h, width w and mass M. Neglect air
resistance.
(a) Find the instantaneous angular velocity of the door about its hinges when it has
swung through 90◦
.
(b) Find the horizontal force on the door when it has swung through 90◦
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a)
1 The truck accelerates forward at a constant rate A, and the door begins to swing
shut. The door is uniform and solid, with height h, width w and mass M. Neglect air
resistance.
(a) Find the instantaneous angular velocity of the door about its hinges when it has
swung through 90◦
.
(b) Find the horizontal force on the door when it has swung through 90◦
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A)
The center of mass of the door is at its geometric center, h/2 from the top or bottom edge, and w/2 from the left or right edge. The moment of inertia is, using the parallel axis theorem, (1/12)Mw² + M(w/2)² = (1/3)Mw².
B)
The truck is uniformly accelerating horizontally, so in a frame moving with the truck, it appears as though there is a horizontal component to gravity with a value of -A. Conservation of energy then gives: MAw/2 = 1/2 Iω² or ω = (3A/w)½.
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