A loaded cement mixer drives onto an old drawbridge, where it stalls with its ce

Question

A loaded cement mixer drives onto an old drawbridge, where it stalls with its center of gravity three-quarters of the way across the span. The truck driver radios for help, sets the handbrake, and waits. Meanwhile, a boat approaches, so the drawbridge is raised by means of a cable attached to its end opposite the hinge (see figure). The drawbridge span is 40.0 m long and has a mass of 11,600 kg; its center of gravity is at its midpoint. The cement mixer, with driver, has a mass of 29,700 kg. When the drawbridge has been raised to an angle of 30 degree above the horizontal, the cable makes an angle of 70 degree with the surface of the bridge. What is the tension T in the cable when the drawbridge is held in this position? What are the horizontal and vertical components of the force the hinge exerts on the span?

Explanation / Answer

a) I'll assume that at that moment, the bridge is stationary.
The torque about the hinge, caused by the bridge itself, is:
= (11600 kg)×(9.8 m/s²)×cos(30)×(40 m)×(1/2) = 1968995.358 Nm

The torque about the hinge, caused by the truck, is:
= (29700 kg)×(9.8 m/s²)×cos(30)×(40 m)×(3/4) = 7561960.621 Nm

The torque about the hinge, caused by the tension in the cable, is:
= T×sin(70)×(40 m) = (37.59 m)×T

You know that
+ =
so
(1968995.358 Nm) + (7561960.621 Nm) = (37.59 m)×T
T = 253550.3054 N < - - - - - - - - - - - - - - - - - - - - - - - - - - tension in the cable

b) i) The only horizontal force acting on the hinge ist the horizontal component of the tension in the cable, so:
Rh = (253550.3054 N) × cos(70-30)
Rh = 194230.8025 N < - - - - - - - - - - - - - - - - - - - - - - - - - horizontal reaction at hinge

b) ii) The vertical component of the reaction force at the hinge is
Rv = (total weight of construction) - (vertical component of tension in cable)
Rv = [(29700 kg) + (11600 kg)]×(9.8 m/s²) - (253550.3054 N)×sin(70-30)
Rv = 241761.0053 N < - - - - - - - - - - - - - - - - - - - - - - - - - vertica reaction at hinge

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