What Forces Act on the Hinge When the Lift Cable Breaks?

[integra2k20]

Homework Statement

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 (Fig. P12.20). Unfortunately his squire lowered the draw bridge too far and finally stopped it 20.0° 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 8.50 m long and has a mass of 2200 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 1000 kg. Suddenly, the lift cable breaks! The hinge between the castle wall and the bridge is frictionless, and the bridge swings freely until it is vertical.

Note: please see image attached for a better picture of what is going on.

I figured out that the angular acceleration (alpha) is 1.625rad/s^2, and that when the bridge hits the side of the castle wall (when it is vertical going down), the angular speed is 1.5 rad/s.

The part I need help with is as follows.

(c) Find the force exerted by the hinge on the bridge immediately after the cable breaks. (R = ________ i + ________j)

(d) Find the force exerted by the hinge on the bridge immediately before it strikes the castle wall. (R= _______ i + _______j)

Homework Equations

Sum of Torque => Ia = mgLcos(20) (where mg is weight of the bridge, I is moment of inertia of a rod, a = alpha = angular acceleration).

The Attempt at a Solution

I tried drawing free-body diagrams for the solution. My teacher said the problem has nothing to do with the weight of the knight and the horse (even though they tell you the weight of the combined mass in the question, supposedly it is irrelevant), so the only forces I can see affecting the bridge are the weight of the bridge itself, and the force that the joint with the wall has on the bridge. I tried solving using the sums of these forces and setting them equal to Ia with no luck. Any help would be greatly appreciated!

 

[andrevdh]

(c) The reaction force on the bridge will be along the direction of the bridge and pointing away from it due to the frictionless hinge. This means that the components of the weight of the bridge and the knight along the bridge need to balance the reaction force.

(d) In this case we have a situation like a swinging pendulum. This means that the reaction force at the hinge need to balance the weight of the bridge (knight has fallen off!) and provide it with the necessary centripetal acceleration.