Tension and slings

*Studiot*

I don't know why we are making this so theoretical, the original question was a practical one from a practical man.

I posed the question about the capstan to demonstrate the power of friction and winding the rope around the beam.

The difference in tension between the two ends of a rope wound round a beam or capstan etc is

T₁ = T₂exp(2n∏μ)

Where T₁ & T₂ are the tensions at each end, n is the number of turns, μ is the coefficient of friction.

To illustrate the power of this say the coefficient of friction is as low as 0.2.

For six turns the tension at one end will be nearly 2000 times the tension at the other. (Yes two thousand)

Consider a cowboy hitching his horse to a beam or rail.
If he takes 6 turns of the reins around the beam and hangs a loose end down weighing just 4 ounces or 100 grams the horse will need to pull 1/5 ton or nearly 450 lbs or 2000N to free himself.

nasu

Possibly because the original question described a rope attached to the supported body with both ends. There was no free end of the rope. At least this what can be seen in both drawings posted on the previous page.

When you mentioned friction, you did not say that you are thinking about someone pulling the free end of a rope, in which case indeed friction is very important (as applied for example to attaching a boat to the poles on the pier).
As you said, an image may help

When you first mentioned friction, you somehow omitted to say that you changed the setup.
I hope my (crude) figure will help to clarify the discussion.

*Studiot*

When you first mentioned friction, you somehow omitted to say that you changed the setup.

Despite many rumours to the contrary I did no such thing.

I specifically outlined and discussed separately two cases, one with a simple passing over the beam, which I likened to a (frictionless) pulley and a second with the case of taking some securing turns around the beam.

If however you take several turns around the beam.............

Chestermiller

Quote by Studiot

I don't know why we are making this so theoretical, the original question was a practical one from a practical man.

I posed the question about the capstan to demonstrate the power of friction and winding the rope around the beam.

The difference in tension between the two ends of a rope wound round a beam or capstan etc is

T₁ = T₂exp(2n∏μ)

Where T₁ & T₂ are the tensions at each end, n is the number of turns, μ is the coefficient of friction.

To illustrate the power of this say the coefficient of friction is as low as 0.2.

For six turns the tension at one end will be nearly 2000 times the tension at the other. (Yes two thousand)

Consider a cowboy hitching his horse to a beam or rail.
If he takes 6 turns of the reins around the beam and hangs a loose end down weighing just 4 ounces or 100 grams the horse will need to pull 1/5 ton or nearly 450 lbs or 2000N to free himself.

I think it would be worth mentioning that the above equation is valid for static friction just at the point where the rope begins to slip, and thus represents an upper bound to the tension ratio when the rope is not slipping. For kinetic friction, the equation is valid at all slip velocities.

nasu

Quote by Studiot

Despite many rumours to the contrary I did no such thing.

I specifically outlined and discussed separately two cases, one with a simple passing over the beam, which I likened to a (frictionless) pulley and a second with the case of taking some securing turns around the beam.

If however you take several turns around the beam.............

I suppose I missed one case from the drawing:

This is how I understood your last sentence. I hope that you agree that it may be taken this way, in the context of previous discussion, unless you specifically mention that one end of the rope is free.

*Studiot*

Certainly that's three cases if you like. I didn't specify what you do with the loose end if there is one, any more than I spent much discussion of the simple (pulley) case since others had already done that to death.

I was really trying to emphasis the importance taking securing turns, something any slinger or rigger should know.

I think I've probably done that to death too now.

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Studiot	Certainly that's three cases if you like. I didn't specify what you do with the loose end if there is one, any more than I spent much discussion of the simple (pulley) case since others had already done that to death. I was really trying to emphasis the importance taking securing turns, something any slinger or rigger should know. I think I've probably done that to death too now.