The Wicked King and the Beautiful Princess - Tension

[Talha Temuri]

I have no idea where to begin. Any help would be appreciated.

Once upon a time there was a terribly wicked King who had a daughter who was very beautiful. The daughter was in love with a handsome prince, and before long the prince and the princess had become inseparable. The King, however, was wicked and did not believe in happiness and so he had his daughter locked up in a prison at the top of a tall tower.

The prince learned of this and was determined to rescue the one he loved, so he started out for the tower where the unhappy princess sat imprisoned. When he arrived at the base of the tower the prince looked up and noticed that there was a wooden beam protruding from the top of the structure. He immediately contrived a method to use this, to reach his princess. He attached a sturdy bucket, he found nearby, to one end of a very long rope and to the other end he tied a stone. Then with a mighty heave he threw the stone across the top of the beam so that the rope was looped across the top of the beam. The prince had thus constructed a simple pulley. He then stepped into the bucket and proceeded to hoist himself up. (See the well-drawn diagram)

In due time the prince reached the top and was rewarded with a long embrace by the King's daughter. The prince, however, could not return the embrace, nor could he begin his work to release the princess, since letting go of the rope would cause the bucket to fall. So he began searching for a way to fix the rope to the tower wall. Luck seemed to be smiling on the young man because close by, he discovered a metal hook embedded in the stonewall. The prince tugged on the hook with one hand (the other hand holding the rope tightly), and finding it secure, he proceeded to tie the rope to the hook. But the instant he did that, the supporting beam broke and the bucket, together with the poor prince, came crashing to the ground. The King, who was very wicked, also happened to know his physics very well (no connection between the two), and had originally designed the beam to support the weight of the prince and the weight of the bucket, but no more How did the King know the prince would fall? Show using diagram and equations.

 

[PeterDonis]

Can you give a reference to where you got this problem from?

 

[PeterDonis]

Also, you need to use the homework template--you need to give the equations that you think might apply and show your own attempt at a solution. Just saying "I have no idea where to begin" is not enough.

 

[haruspex]

I have no idea where to begin.

Yet forum rules require you to show some attempt.
Start with two diagrams, one showing the prince, bucket, rope and beam before he tied the rope to the hook, the other after he has tied it and let go the rope.
In each, show the forces on the rope, on the prince+bucket, and on the beam.
Post your diagrams, and any equations you can develop.

 

[kuruman]

I agree that the template should be followed, but I am not sure equations are needed. The king had

... originally designed the beam to support the weight of the prince and the weight of the bucket, but no more.

Just look at the picture. There is more than just the prince and the bucket producing tension in the rope.

 

[J Hann]

It seems to me that I read that Gauss had an argument with his teacher about a similar problem.
"if a person sitting on a bench with a rope attached to the bench and passing over a pulley tried
to raise himself by pulling on the other end of the rope, what force would be required?"
Apparently, he went home and tested this out in a barn.
The teacher's answer was W and Gauss' answer was W/2.

 

[kuruman]

Raising generally implies acceleration unless done at constant speed. I would rephrase the question to ask "If a person raised himself above the bench at some fixed height, with what force does he need to pull on the rope to stay at that height? Then Gauss's answer would be correct.

 

[J Hann]

Even if the force required is somewhat greater than W/2 and the force on the pulley (beam) is somewhat greater
than W, what is the force on the pulley (beam) when the rope is attached to the hook and the prince let's go of the rope?

 

[kuruman]

Even if the force required is somewhat greater than W/2 and the force on the pulley (beam) is somewhat greater
than W, what is the force on the pulley (beam) when the rope is attached to the hook and the prince let's go of the rope?

You have two ropes each puling down with tension T on the beam. As long as the rope is under tension, the force on the beam is 2T. This can be idealized as an Atwood's machine situation with a well known answer for the tension T. On one side you have W₁ = W_prince + W_bucket and on the other side you have W₂ = W_stone.

 

[haruspex]

@kuruman, @J Hann , I'm not sure what you are debating in this thread but you are in danger of laying out the solution for the OP (who has not resurfaced since post #1), or of obscuring it with irrelevancies.

The acceleration can be made effectively irrelevant by making it arbitrarily small.

The Gauss anecdote concerns the tension in the rope rather than the force on the beam. It might be helpful, or it might be confusing. We should wait and see where the OP is stuck.

As I read the question, the stone was only used to get the rope over the beam and now lies harmlessly on the ground.

 

[PeterDonis]

As I read the question, the stone was only used to get the rope over the beam and now lies harmlessly on the ground.

That would make sense with the description in words, but it is not consistent with the picture given in the OP, which shows the stone hanging in the air as the prince pulls himself up. Which might just mean the picture and the words are inconsistent and whoever put the problem together didn't notice. Since we still don't know where this comes from, we don't know how reliable the source is.

 

[PeterDonis]

We should wait and see where the OP is stuck.

To make that easier, I have partially locked down the thread awaiting a further post from the OP.

 

[haruspex]

That would make sense with the description in words, but it is not consistent with the picture given in the OP,

You are right, the illustration violates this statement:

weight of the prince and the weight of the bucket, but no more