How Is Torque Calculated in a Drawbridge Pulley System?

[electerr]

Homework Statement

A drawbridge to the castle is slowly lifted. Its weight is 500kg. There are is one chain drawn over a pulley that pulls up the drawbridge (see pic). The angle is 30 degrees at the rotation point (see pic).

http://pici.se/200459/

link to pic if it dosen't come up:
http://pici.se/200459/

Calculate the force in each section of chain as the picture shows.
Decide the scenerio that puts the chain under the greatest force and calculate that force.

Homework Equations

f1 * L1 = f2 * L2

The Attempt at a Solution

The first question I get but the other I can't figure out for the life of me.. Tips?

 

[Q_Goest]

Hi electerr.
The first question may be improperly translated.

Calculate the force in each section of chain as the picture shows.

I'm going to assume this should read something like, "Calculate the force in the chain for the position shown in the picture." It could be however that you have 2 chains that are identical, so you have to take the total force and divide by 2. Let's resolve the forces assuming 1 chain and if the problem says there are 2 identical chains, just divide the force by 2.

Do you know how to take moments around a point? Assume that the sum of the moments around the hinge of the drawbridge are equal to zero (assume the static equilibrium condition). Can you write the equation for the moment the chain creates around the hinge? Can you write the equation for the moment around the hinge that the drawbridge creates? Write these two out. Use variables because you will reuse these equations again for the second part of the problem.

Once you write the equations out, you should know that the sum of these two equations is equal to zero for a bridge in static equilibrium. Equate them and solve for tension in the chain.

The second question :

Decide the scenerio that puts the chain under the greatest force and calculate that force.

follows from the first question. You should be able to create a single equation that determines the tension in the chain as a function of the angle the drawbridge makes. The angle that maximizes the chain tension is the "scenario that puts the chain under the greatest force". If you have an equation for tension in the chain, T, which is a function of the angle the bridge makes, do you know how to determine the maximum possible T? Don't forget that the angle is limited to a maximum of 90 degrees.

 

[electerr]

Yeah, your probably right about the translation... the problem is translated from swedish and my physics termonology in english isn't so accurate. Yes, the chain is divided in two as you said and yes, you solve the problem dividing the total force by 2.

F * L = 4910N * (L/2) * sin(30)

I simplify L from the equation and get 1227.5N total force on the chain.

It is the 2nd ? that causes me the problems...
I know that the most force generated when lifting the bridge will be when it is 90 degrees to the wall but because I have no real measurments to put in the equation then I'm guessing that I just have to represent the measurments as I did in the first problem but I can't seem to get my head around how to do that...

 

[Q_Goest]

Hi electerr. I'm at work today so I can't spend much time on this. But what measurement is L in your equation? Is that the length of the drawbridge?

You should be able to do the same thing for the second question. Note the chain won't be perpendicular to the bridge except for the position shown, so you may need to come up with a more generic equation for the moment around the hinge caused by the chain. But if you already 'know' the maximum tension is when the drawbridge is horizontal, you can just do the equation for moment arm to the chain making that assumption. In other words, the moment arm is at an angle of 45 degrees from horizontal.

 

[timmay]

electerr:

If you suggest that the tension is greatest in the chain at horizontal, perhaps a diagram will help you. If you look at my attached image, I've suggested the simplest ways of resolving the forces for both the first part of the problem and the second, but instead of specifically listing the angles, I've used the variables alpha and beta.

To resolve the force in the second case, you're best off showing what angle beta is first. Looking at the triangle that the drawbridge makes with the wall, you immediately see that there are two fixed points for the system - the pivot point, and the point at which the cable is attached to its pulley. Thus you have two lengths in the system: the length of the bridge, and the vertical distance to the pulley. Using trigonometry you should then be able to describe how the angle beta changes as the angle alpha changes.

Once you've done that, you should be able to use trigonometry to resolve the tension in the chain (by balancing the moments as you've already done) for any angle alpha and its resulting angle beta. From that, you should either be able to check your hypothesis for alpha equals 90^o or if you were really dedicated you could find the angle for which the expression for the resultant is a maximum by differentiation.

Hope this helps.

 

[electerr]

Yes! That worked. Thanks