Mechanical Advantage Question - Ropes

[Mdanner423]

I have to take an aptitude test that I have taken before and there are a couple questions that are stumping me.

My first question I want to ask about deals with ropes / strings. It has an old-timey picture of a boy carrying books wrapped up in string and then attached to a stick and thrown over his back. One picture the string/rope is wrapped around the stick more. It ask which is easier to carry, A (the rope is wrapped around twice), or B (the rope is wrapped around five times), or C it's equal.

My first guess is B, but I can't explain why.

So does wrapping the rope more around help distribute the mass anymore to make it easier to carry or?

Thanks!

--Matthew

 

[emailanmol]

Unfortunately, its the opposite.

At higher level of physics , you can derive the expression of the force on the stick by rope which increases exponentially with an increase in the number of rotations .

So the boy will have to provide more force to hold the stick if it has five rotations.

 

[Mdanner423]

Interesting! Can you explain why?

 

[emailanmol]

And yes, my previous post is valid in presence of friction.
If friction is absent the number of turns doesn't matter :-)

 

[emailanmol]

Surely I can,

Consider a small piece of the rope that subtends an angle dθ. Let the tension in this piece be T (which will vary slightly over the small length). the pole exerts a small outward normal force, N, on the piece. This normal force exists to balance the inward components of the tensions at the ends. These inward components have magnitude T sin(dθ/2). Therefore, N= 2T sin(dθ/2). The small-angle approximation, sin x ≈ x, then allows us to write this as N= T dθ.
The friction force on the little piece of rope satisfies Fdθ ≤ μN= μT dθ. This friction force is what gives rise to the difference in tension between the two ends of the piece. In other words, the tension, as a function of θ, satisfies


T(θ+dθ) ≤ T(θ)+μTdθ
=⇒ dT≤μTdθ
=⇒ dT/T≤μdθ
Integrating both sides

=⇒ ln T ≤ μθ + C
=⇒ T ≤ T(0)e^μθ,
where T =T(0) whenθ=0.


So when theta is 4pi (720 degrees)which correponds to two turns tension is large. As theta increases this gets huge.


In laymans terms this is happening because of friction.
Since more part of rope is in contact with stick when rotations are more,more friction is acting for same smaller movement.


(This is a classic example whicg is given in DAVID MORINS Classical Mechanics)

 

[A.T.]

So the boy will have to provide more force to hold the stick if it has five rotations.

I'm not sure if I picture this correctly.

===\\\\=== STICK
   |
   | ROPE
   |

Do you claim that for the same force pulling down at the rope the force needed to hold the stick depends on the number of windings? (Assuming the same total length and thus mass of the rope)

 

[Mdanner423]

Same total length and mass, does the force required change depending on how many times the rope is wound around?

 

[emailanmol]

I'm not sure if I picture this correctly.

===\\\\=== STICK
   |
   | ROPE
   |

Do you claim that for the same force pulling down at the rope the force needed to hold the stick depends on the number of windings? (Assuming the same total length and thus mass of the rope)

Hey A.T. ,

Yess it does because of friction(if friction is absent, then it doesn't depend) :-)

Friction is proportional to the number of turns as more part of rope (for same length and same mass) is in contact with the stick when no. of turns is higher.
(picture it this way, if more part is in contact, more part of rope has chances of having relative motion wrt to the stick.So friction force applied will be more )

The net frictional force applied by the stick on rope is upwards (as it wants to prevent the ropes from moving down with the books) so the force applied by rope on stick is downwards.

We therefore need to apply more force to hold the stick to balance this extra force of friction.

 

[Mdanner423]

So as you wrap it there isn't a proportional upward force making it equal? I appreciate your answer I just don't understand it!

 

[emailanmol]

So as you wrap it there isn't a proportional upward force making it equal? I appreciate your answer I just don't understand it!

Hey,
No issues.


Look at it this way.
The book is pulling the ropes down with mg(m is mass of books) and the stick is trying to resist this by friction.

(No other force acts on rope except its own weight which also acts downwards).

So the net frictional force on rope is upwards and towards the man )(downwards on stick and away from man)

If more part of rope is in contact (and suddenly a small relative motion between the ropes and stick takes place) with the stick, the stick applies friction on much more portion of string (as a large portion is having relative motion wrt stick)
and therefore net friction is higher.

Think of it this way.
If a block was lying on two tables with half the surface area on one and half the surface area on two.
If one table was smooth and the other had friction, less force of friction will act as compared to situation where both tables were rough. :-)

Right??

 

[Mdanner423]

Okay increasing the loops of rope wrapped around the stick increases the friction between the stick and the rope. But, how does this make it more difficult to raise against gravity in the Y direction?

 

[A.T.]

If more part of rope is in contact (and suddenly a small relative motion between the ropes and stick takes place) with the stick, the stick applies friction on much more portion of string (as a large portion is having relative motion wrt stick)
and therefore net friction is higher.

Are we talking about the static case or a rope that is sliding down?

 

[Mdanner423]

Static, the rope shouldn't be going up and down, but obviously would sway as you walk with it.

Here's another way to phrase the question. Let's say you take a fishing pole with a lure on the end, does it take more or less force to lift the lure if you wrap the fishing line around the pole?

 

[M Quack]

Okay increasing the loops of rope wrapped around the stick increases the friction between the stick and the rope. But, how does this make it more difficult to raise against gravity in the Y direction?

No, of course not. As long as the situation is static, i.e. the rope is not sliding it makes absolutely NO difference how it is attached to the stick. By a nail, a few turns, many turns, a knot or duct tape. You name it, it makes no difference at all. The force on the stick is always the weight of the books (plus string, nail, duct tape, etc.). If you tie the knot more tightly, you will compress the stick more, but for carrying it that makes no difference unless you break the stick while doing so.

The package might be more comfortable to carry if the free length of the rope, i.e. the distance between the stick and the books, is shorter, because the books will swing back and forth less.

 

[emailanmol]

Okay increasing the loops of rope wrapped around the stick increases the friction between the stick and the rope. But, how does this make it more difficult to raise against gravity in the Y direction?

Hey,
See we hold the stick by balancing all the forces acting on it.

Forces acting on stick are its weight mg (down) and force of friction(down).
So we need to apply an upward force to balance it.


Its v imp to understand that the net frictional force on rope by the stick acts along the direction of man and in the upward direction i.e at an oblique angle facing upwards.(This is the vector sum of all those small tiny forces acting tangentially on the ring of rope)

This is reasoned by logic.

look at the free body diagram of rope.
Its weight + force applied by books act downwards.
And the only other force acting on it is friction.

Now the rope moves with the man along x-axis so the only force which moves it is friction.also its this friction which balance the weight of books and rope .

Are you understanding?

I hope I am making it ckear enough :-)

 

[M Quack]

You are forgetting your boundary conditions, which are F=0 at the free end of the rope, and F=mg at the end going towards the books, independent of the number of turns.

Having more turns just means that you "distribute" the same friction force over more turns and more length of rope. The rope will be wound more loosly (many turns) or more tightly (few turns), which decreases or increases the friction.

 

[emailanmol]

No, of course not. As long as the situation is static, i.e. the rope is not sliding it makes absolutely NO difference how it is attached to the stick. By a nail, a few turns, many turns, a knot or duct tape. You name it, it makes no difference at all. The force on the stick is always the weight of the books (plus string, nail, duct tape, etc.). If you tie the knot more tightly, you will compress the stick more, but for carrying it that makes no difference unless you break the stick while doing so.

The package might be more comfortable to carry if the free length of the rope, i.e. the distance between the stick and the books, is shorter, because the books will swing back and forth less.

Everything you said is perfectly right.However, i I tend to disagree on few points on basis of practicality and not theoretically .

the rope won't practically remain static with respect to the stick always. In case, even a little motion takes place (which is bound to practically depending upon the jerks in motion ) the man will have to suddenly apply a larger force to balance the stick and the stick can even break.

And making more rotations around the rope won't decrease the length a lot. Its a thin stick .

 

[emailanmol]

Strictly speaking, nothing can be said about the perfect number of rotations on stick.

It is advisable to have a large number of rotations around the package as not only will this create a friction to precent slipping it will also help decrease the pressure on rope which can cause stress and strain and break it.

However number of rotations on stick will depend upon the value of u, the strength of stick and man, the weight of books etc.
Because we want number of rotations such that the force of static friction is sufficient to prevent small movements but also isn't large enough that if movement takes place
the stick breaks (or the man fails to lift it)

However, in general number of rotations around stick should be as few as possible.

 

[emailanmol]

Static, the rope shouldn't be going up and down, but obviously would sway as you walk with it.

Here's another way to phrase the question. Let's say you take a fishing pole with a lure on the end, does it take more or less force to lift the lure if you wrap the fishing line around the pole?

As M Quack rightly suggested
If its static wrt stick, it doesn't make a difference how many number of loops are present.

So force would be same.

 

[M Quack]

What you have calculated very nicely is the force needed to pull the rope off the stick.

The question was how "easy" it is to carry the stick and package. Unless this is a rather nasty trick question, the force concerned is the force of gravity on the stick under static conditions. In this (admittedly simplified and restricted) case the force to pull the string off is irrelevant.

 

[emailanmol]

You are forgetting your boundary conditions, which are F=0 at the free end of the rope, and F=mg at the end going towards the books, independent of the number of turns.

Having more turns just means that you "distribute" the same friction force over more turns and more length of rope. The rope will be wound more loosly (many turns) or more tightly (few turns), which decreases or increases the friction.

What you have calculated very nicely is the force needed to pull the rope off the stick.

The question was how "easy" it is to carry the stick and package. Unless this is a rather nasty trick question, the force concerned is the force of gravity on the stick under static conditions. In this (admittedly simplified and restricted) case the force to pull the string off is irrelevant.

Hey,
I coudn't agree more. :-)
You are right but isn't this valid only during static condition.(I agree on whatever you have said uptil now, but only for static conditions.)

When the rope is not static, larger friction acts for more rotations which makes it tougher for man.

 

[M Quack]

If the package was nailed to the ground, more turns make it more difficult for the guy to get away :-)

In a practical situation, I would want the situation to be static, i.e. more turns would be better because the threshold for sliding would be higher. But in a practical situation I would make a knot anyways.

 

[Mdanner423]

So in summary... It would be easier to carry with more turns as it would make it more stable, sway less back and forth, and make it harder for the rope to slide off the stick. It takes equal force (N) to lift the books against gravity regardless of the number of turns. Right?!

 

[M Quack]

Right.

 

[emailanmol]

So in summary... It would be easier to carry with more turns as it would make it more stable, sway less back and forth, and make it harder for the rope to slide off the stick. It takes equal force (N) to lift the books against gravity regardless of the number of turns. Right?!

Hey, i still would like to add an additional point as I tend to diagree (a bit)with the answer.(in good spirit,offcourse !:-) )


Suppose you had n turns on your stick at this moment.(in the situation you are talking about like no swaying...etc)

If you decide to Now
make n+1 turns , it would worsen the situation and not improve it .


Because in case any relative motion takes place(and i can guarantee you, practically it will not take place for both 2 or 5 turns but if does 5 turns is a burden rather than relief)

the force applied by us is higher in case of n+1 turns.(and the stick might break rendering the situation even more useless)

In essence having a larger number of turns increases the threshold of static friction thus decreasing probability of motion, but it also adds a factor of huge instability which equation T=T(0)e^u(theta) indicates.

If you take u = 1/2( a very practical value),then two rotations increases the force factor by 530 times, and four by (theta is 8pi)300,000 times.

You can think of how much effect 5 rotations would cause (in fact you can calculate).

Practically even few rotations (like 1 or two for a normal stick and string)will take care of all the factors you are worried about(threshold static friction...etc) for a significant portion of the journey and adding more turns is hurting your chances rather than improving it
because if a slight amount of relative motion (though unlikely but still) takes place the stick will break or you won't be able to hold it