Solving the Mechanics Puzzle of Meshcherskii: Understanding Thrust and Drag

[cwill53]

Homework Statement

5. Three barges of different displacement are being towed one behind the other by a tugboat. The thrust developed by the propeller of the tug is 3600lbs, the drag of the tug is 1200lbs. The drags of the first, second, and third barge are 1200lbs, 800lbs, and 400lbs, respectively. The tensile strength of the available rope is 400lbs. How many ropes of this strength must be used to attach each barge to the preceding one? The motion of the train is rectilinear and uniform.

Relevant Equations

$$\sum \vec{F}=m\vec{a}$$

"Collection of Problems of Mechanics" by Meshcherskii.I see how the answer could be arrived at because each successive drag increases the number of ropes that must be used, but what I don't understand is why the thrust and drag of the tug doesn't play a role in the number of ropes required.

 

[etotheipi]

Have you drawn a diagram?

 

[cwill53]

Have you drawn a diagram?

I did but I don't think it's necessary to use one to solve this problem.

 

[etotheipi]

I did but I don't think it's necessary to use one to solve this problem.

Is it possible to post it here, or at least describe all of the forces on each object? A clear diagram is often helpful to avoid errors in questions like these. And especially useful if you get stuck!

 

[cwill53]

Here is my diagram which may been inaccurate bc the forces may be in the wrong order.

 

[etotheipi]

Your diagram is fairly good (I like the boat!), but there are one or two things you could do better. Firstly making sure the force arrows are attached to the body they're acting on (might sound pedantic, but is a good habit!).

More importantly, you haven't really drawn in the tension forces. A rope will exert a tension force on anything attached to it, so really for each connection you should have two opposite forces a bit like this (ignore the forces acting on the rope):

Once you've sorted that, apply the condition of zero acceleration to different systems of choice to work out your tension forces!

 

[cwill53]

Your diagram is fairly good (I like the boat!), but there are one or two things you could do better. Firstly making sure the force arrows are attached to the body they're acting on (might sound pedantic, but is a good habit!).

More importantly, you haven't really drawn in the tension forces. A rope will exert a tension force on anything attached to it, so really for each connection you should have two opposite forces a bit like this (ignore the forces acting on the rope):

View attachment 261282
Once you've sorted that, apply the condition of zero acceleration to different systems of choice to work out your tension forces!

Lol thanks, I tried with the boat and I will fix up my diagram.

The book says that the answers are 6,3, and 1 rope, but I don’t see how those answers could be arrived at looking at my diagram. I do see how one could get that from

400lbs/400lbs=1
(400lbs+800lbs)/400lbs=3
(400lbs+800lbs+1200lbs)/400lbs=6

I simply don’t understand how the thrust of the propeller doesn’t play a role in this.

 

[haruspex]

what I don't understand is why the thrust and drag of the tug doesn't play a role in the number of ropes required.

It would if the system is accelerating. Is it?

 

[etotheipi]

The book says that the answers are 6,3, and 1 rope, but I don’t see how those answers could be arrived at looking at my diagram.

Okay, first consider the third boat bobbing along on the end of the chain. What's the tension force in that final connection? So how many ropes do you need? Then, consider a system containing the last two boats and the connection between them. What's the tension force in the second connection?

What system do you use to get the third tension?

@haruspex is right, if the system is accelerating you'd need to do a little bit more work.

 

[cwill53]

Your diagram is fairly good (I like the boat!), but there are one or two things you could do better. Firstly making sure the force arrows are attached to the body they're acting on (might sound pedantic, but is a good habit!).

More importantly, you haven't really drawn in the tension forces. A rope will exert a tension force on anything attached to it, so really for each connection you should have two opposite forces a bit like this (ignore the forces acting on the rope):

View attachment 261282
Once you've sorted that, apply the condition of zero acceleration to different systems of choice to work out your tension forces!

It would if the system is accelerating. Is it?

I see now. It all makes sense. Thank both of you for your help.

 

[cwill53]

Okay, first consider the third boat bobbing along on the end of the chain. What's the tension force in that final connection? So how many ropes do you need? Then, consider a system containing the last two boats and the connection between them. What's the tension force in the second connection?

What system do you use to get the third tension?

@haruspex is right, if the system is accelerating you'd need to do a little bit more work.

The tension force in that one would just be 400lbs, right?

What if I tweaked the problem some and said that the system was accelerating. Only then would the thrust and drag of the tugboat matter, right?

 

[etotheipi]

The tension force in that one would just be 400lbs, right?

What if I tweaked the problem some and said that the system was accelerating. Only then would the thrust and drag of the tugboat matter, right?

I mean you can always just call it $a$ and work through the problem again to see what happens! All the boats are constrained to have the same acceleration (so long as it's positive - ropes can't transmit thrust!), so the only real difference it makes is what you set equal to the resultant force when you resolve forces on any chosen sub-system. It's not going to be zero any more...

 

[cwill53]

Came across this question:
A load Q of 60lbs is held in equilibrium by a counterbalance P attached to the end of a rope ABC weighing 10lbs which passes around a pulley. Determine the weight of the counterbalance P, the tensions $$F_{A}$$ and $$F_{C}$$ in the end sections, and the tension in the middle section B of the rope, if (1) the ends A and C are at the same level; (2) the end A is at the highest position; (3) the end A is at the lowest position. Neglect the stiffness of the rope, the radius of the pulley, and friction.

The answers and diagram are in the following photo.

I understand how they might have got to these answers but I'm still kind of shaky on how the mass of the rope plays a role in the tension at point B, and how to mathematically represent the tension at any point along the rope; I know the tension varies because the rope has mass. If I was to consider friction, what would should be done differently?
For part 1, I know intuitively that Q and P have to be the same weight because they're at the same level and the system is in equilibrium. I also know that $$\vec{F_{A}}=\vec{F_{_{C}}}=60lbs(-\hat{y})$$ because the weights Q and P are not holding up the weight of the rope.

 

[etotheipi]

Since it's a different question entirely, you might be better off posting this in a new thread.

With that said, a comment I will add is that you should consider the pulley as being negligibly small, and you should think of the rope as broken down into two solid bodies held together by a connection over the pulley. You can then imagine a tiny massless string between each of the masses and the solid sections of rope, exerting the tension force between them:

 

[cwill53]

Since it's a different question entirely, you might be better off posting this in a new thread.

With that said, a comment I will add is that you should consider the pulley as being negligibly small, and you should think of the rope as broken down into two solid bodies held together by a connection over the pulley. You can then imagine a tiny massless string between each of the masses and the solid sections of rope, exerting the tension force between them:

View attachment 261342

Thanks. Do you think it would be appropriate to keep related questions in the same thread? I’m working on changing my study habits and doing a lot more problems and I know that I will have more questions.

 

[etotheipi]

Thanks. Do you think it would be appropriate to keep related questions in the same thread? I’m working on changing my study habits and doing a lot more problems and I know that I will have more questions.

I would advise opening a new thread for a completely new question, it's also beneficial for you since it means more people will see the new post and be able to help.

 

[cwill53]

I would advise opening a new thread for a completely new question, it's also beneficial for you since it means more people will see the new post and be able to help.

Will do