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 11,900 kg; its center of gravity is at its midpoint. The cement mixer, with driver, has a mass of 28,100 kg. When the drawbridge has been raised to an angle of 30

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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