Sir Lost-a-Lot dons his armor and sets out from the castle on his trusty steed i

Question

Sir Lost-a-Lot dons his armor and sets out from the castle on his trusty steed in his quest to improve communication between damsels and dragons. Unfortunately his squire lowered the draw bridge too far and finally stopped it 20.0 degree below the horizontal. Lost-a-Lot and his horse stop when their combined center of mass is 1.00 m from the end of the bridge. The uniform bridge is 9.00 m long and has a mass of 1900 kg. The lift cable is attached to the bridge 5.00 m from the hinge at the castle end, and to a point on the castle wall 12.0 m above the bridge. Lost-a-Lot's mass combined with his armor and steed is 1050 kg. Determine the tension in the cable. Determine the horizontal force component acting on the bridge at the hinge. magnitude direction to the left to the right Determine the vertical force component acting on the bridge at the hinge. magnitude direction upwards downwards

Explanation / Answer

first we calculaTE the angle the cable makes with the bridge.

The angle between the bridge and the wall (vertical) is 90-20 = 70º.
The law of cosines gives us
C² = 5² + 12² - 2*5*12*cos70º = 128
C = 11.3 m (length of cable)

Now we can use the law of sines to get the angle between the bridge and cable:
sin / 12m = sin70º / 11.3m
sin = 0.997
= arcsin0.997 = 85.46º

(a) Now sum the moments about the hinge:
M = 0 = (1900kg * ½*9m + 1050kg * 8m)*9.8m/s²*cos20º - T*sin85.46º*5m
where T is the cable tension.
Solving, find
T = 156092.34 N·m / (5m*sin85.46) = 31316 N

c) downwards

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