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.

 

[m2003]

I would approach this problem by first understanding the concept of torque and its relationship to force and angular acceleration. In this scenario, we have a bridge that is being supported by a hinge and a lift cable, and the bridge is also subject to the gravitational force due to its own weight.

To find the force exerted by the hinge on the bridge immediately after the cable breaks, we can use the equation for torque (τ = r x F) and the given information about the bridge's moment of inertia (I), angular acceleration (α), and angle of inclination (20°). We can also use the fact that the bridge will rotate around the hinge when the cable breaks, so the torque acting on the bridge will be equal to the moment of inertia multiplied by the angular acceleration (τ = Iα).

Using this information, we can set up the equation τ = Iα and solve for the force exerted by the hinge (F). The equation would be F = τ/r = Iα/r, where r is the distance from the hinge to the center of mass of the bridge.

For part (c), where we are looking for the force exerted by the hinge immediately after the cable breaks, we can use the given information about the bridge's length, mass, and the distance from the hinge to the center of mass to solve for the force. The answer would be in the form of a vector, with components in the i and j directions.

For part (d), where we are looking for the force exerted by the hinge immediately before the bridge strikes the castle wall, we need to consider the fact that the bridge will have a different angular acceleration at this point (since it will be hitting the wall), and we also need to take into account the force of gravity acting on the bridge. We can use the same equation (F = τ/r) but we need to consider the total torque acting on the bridge, which would be the torque due to the force from the hinge and the torque due to the force of gravity. We can set up an equation for the total torque and solve for the force from the hinge.

In summary, to solve this problem, we need to use the concepts of torque, force, and angular acceleration, and apply them to the given scenario. We also need to consider the different stages of motion of the bridge (when the cable breaks and when it hits the castle wall) and the different forces acting on the bridge