Sir Lost-a-Lot dons his armor and sets out from the castle on his trusty steed (
ID: 1472882 • Letter: S
Question
Sir Lost-a-Lot dons his armor and sets out from the castle on his trusty steed (see figure below). Usually, the drawbridge is lowed to a horizontal position so that the end of the bridge rests on the stone ledge. Unfortunately Lost-a-Lot's squire didn't lower the drawbridge far enough and stopped it at ? = 20.0° above the horizontal. The knight and his horse stop when their combined center of mass is d = 1.00 m from the end of the bridge. The uniform bridge is ? = 8.00 m long and has a mass of 2000 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 h = 12.0 m above the bridge. Lost-a-Lot's mass combined with his armor and steed is 1000 kg.
(a) Determine the tension in the cable.
(b) Determine the horizontal force component acting on the bridge at the hinge.
magnitude N
direction
(c) Determine the vertical force component acting on the bridge at the hinge.
magnitude N
direction
Explanation / Answer
We need the angle the cable makes with the bridge.
The angle between the bridge and the wall (vertical) is 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
The weight vector for the bridge is located at the center of the bridge. This point is 4 meters from the hinge. The weight vector for the man and horse is located at the 1 meter from the end of the bridge.
(8 – 1 = 7) This point is 7meters from the pivot point.
The lift cable is attached to the bridge 7.00 m from the hinge
(a) Now sum the moments about the hinge:
M = 0 = (2000kg * ½*8m + 1000kg * 7m)*9.8m/s²*cos20º - T*sin85.46º*5m
where T is the cable tension. Solving, find
T = 138134.8N·m / (5m*sin85.46) = 27713.92 N 27714 N
(b) ß = 180º - 70º - 85.46º = 24.54º angle between cable and wall
horizontal force Fx = T*sin24.54 = 11510.37 N
away from wall
(c) vertical force Fy = (2000 + 1000)kg * 9.8m/s² - T * cos24.54 = 32319.35 N
up