How Does a Man Accelerate While Climbing a Rope?

[Kaushik]

TL;DR Summary

I am not able to understand what happens when a man climbs up a rope.

A man tries to climb up a rope with acceleration, $a$. What does he actually do to climb up?

My Interpretation
Let the man pull the rope at point A. So the Point A will pull the man with Tension, $T$. But at the same time the man is holding the rope, so there will be some normal reaction due to which there will be friction acting on the man in the upwards direction. So Is it $T$ and the Friction which pulls him up?
Is this friction static or kinetic? I first thought that it was kinetic, but at that instant the man's hand is pulling the rope but it is not in motion so static friction must act on it.

Is this the actual scenario? If not, what actually is happening? I am confused.

Thanks.

 

[.Scott]

This question is a bit out of context. I'm guessing they are looking for a formula.
How does g (the acceleration due to gravity) fit into this?

 

[Kaushik]

This question is a bit out of context. I'm guessing they are looking for a formula.
How does g (the acceleration due to gravity) fit into this?

I asked this to clarify my doubt. Like I can't get the intuition of what actually happens when a man tries to climb up.

 

[anorlunda]

For the hand to not slip on the rope, there must be a static friction force. It is complicated with a stranded rope because the is not cylindrical in shape.

It is irrelevant what the force of the hand pulling accomplishes.

 

[Kaushik]

For the hand to not slip on the rope, there must be a static friction force. It is complicated with a stranded rope because the is not cylindrical in shape.

It is irrelevant what the force of the hand pulling accomplishes.

So, static friction just prevents the hand from slipping. Then what is the force that makes the $F_{net} > 0$ i.e, in the upward direction? Is it Tension ?

 

[anorlunda]

It sounds like you are confused about the names of the forces.

A man climbing a rope is like a man doing chin ups on a horizontal bar. The forces in his shoulders, arms, elbows and wrists all contribute. Human physiology is very complicated.

Can you restate your question without a human body? Perhaps a weight hanging on a string.

 

[Kaushik]

It sounds like you are confused about the names of the forces.

A man climbing a rope is like a man doing chin ups on a horizontal bar. The forces in his shoulders, arms, elbows and wrists all contribute. Human physiology is very complicated.

Can you restate your question without a human body? Perhaps a weight hanging on a string.

Can we just discuss about what happens around our hand (The forces acting on it and by it)?

 

[jbriggs444]

So, static friction just prevents the hand from slipping. Then what is the force that makes the $F_{net} > 0$ i.e, in the upward direction? Is it Tension ?

Draw up a free body diagram for a small section of rope where the hand grasps on. What forces can you identify that act on that section of rope?

 

[Kaushik]

What forces can you identify that act on that section of rope?

I drew the FBD and noticed that the forces acting on that section of the rope are $F$ with which the man pulls the rope down, $T$ from the supporting rope to keep the rope attached.

On man's hand, I noticed the following forces- $Weight$, $F_{static}$ to prevent the hand from slipping. I am not 100% sure but i feel like the rope also plays its role in preventing the man from falling by pulling it up with same $T$.

 

[jbriggs444]

I drew the FBD and noticed that the forces acting on that section of the rope are $F$ with which the man pulls the rope down, $T$ from the supporting rope to keep the rope attached.

Yes. This is good. I had in mind two additional forces.

$T_{below}$: The tension in the section of rope below the man's hand
$-mg$: The weight of the small section of rope.

For an ideal massless rope, both additional forces are negligible.

On man's hand, I noticed the following forces- $Weight$, $F_{static}$ to prevent the hand from slipping. I am not 100% sure but i feel like the rope also plays its role in preventing the man from falling by pulling it up with same $T$.

You've already counted the force of the rope on the hand as $F_{static}$. You do not get to count it again as tension.

 

[Kaushik]

You've already counted the force of the rope on the hand as FstaticFstaticF_{static}. You do not get to count it again as tension.

This is exactly where I am getting confused. Why don't we count Tension? How does the frictional force include the tension of the string?

 

[.Scott]

Clearly, the real problem is deciphering what the question is trying to address. There is no indication of scope.
Is it sufficient to simply say that the climber is lifting with a force of $<your formula>$? Is friction worth mentioning?

 

[jbriggs444]

This is exactly where I am getting confused. Why don't we count Tension? How does the frictional force include the tension of the string?

Tension acts within the string. It is the force between one section of the string and the next. In a more advanced context, one might not consider it to be a force at all, but something more like a condition (a component of the stress tensor).

Within the string you have tension. Outside the string you have friction. At the interface between the two you can equate the tension inside with the frictional force outside.

The free body diagram is what let's you equate friction and tension. If the chunk of rope where the climber hangs on has negligible mass and acceleration then the supporting force on that chunk (tension) must be equal to the external force that it encounters (friction).

 

[Chestermiller]

Clearly, the real problem is deciphering what the question is trying to address. There is no indication of scope.
Is it sufficient to simply say that the climber is lifting with a force of $<your formula>$? Is friction worth mentioning?

Only the upward (static) friction force of the rope acts on the man's hand, not the rope tension. But, from a force balance on the section of rope spanned by the man's hand, the rope tension above the hand balances the downward friction force exerted by his hand on the rope. So the friction force is equal to the rope tension.

 

[metastable]

Could it be that no work is actually done between the climber’s hand and the rope? Is the real work being done between the climbers’s forearm which has effectively become “coupled” with the rope, and the climber’s upper arm, via the elbow? The upper arm “climbs” in relation to the forearm, which doesn’t climb? Part of the climber’s body climbs, another part stays stationary (until the climber grabs a different part of the rope with their other hand)?

 

[.Scott]

Could it be that no work is actually done between the climber’s hand and the rope? Is the real work being done between the climbers’s forearm which has effectively become “coupled” with the rope, and the climber’s upper arm, via the elbow? The upper arm “climbs” in relation to the forearm, which doesn’t climb? Part of the climber’s body climbs, another part stays stationary (until the climber grabs a different part of the rope with their other hand)?

Yes, all of the work is done by the muscles.

 

[.Scott]

A man tries to climb up a rope with acceleration, $a$. What does he actually do to climb up?

We're not supposed to post flat-out answers, but this thread is going in a very different direction than where I would have gone.

My answer to this question would have been:
The man applies a downward force of $m_m(g+a)$ to the rope.
Where:
$m_m$ is the mass of the man;
$g$ is acceleration due to gravity; and
$a$ is the acceleration relative to the rope.

I would not have gone into the mechanics of climbing.

But what is the context and scope of the question? Who (or what) is the original author of the question. Is this a homework problem? If so, what lesson material is it associated with? What other question are on the homework paper?

 

[metastable]

The man applies a force of mm(g+a)mm(g+a)m_m(g+a) to the rope.

I would not have gone into the mechanics of climbing.

I thought the climber applies the force to their forearm, (since the hand doesn’t move relative to the rope).

 

[.Scott]

I thought he applies the force to his upper arm, (since the hand doesn’t move relative to the rope).

His upper arm is part of "he".
But it doesn't matter how he applies the downward force $m_m(g+a)$ to the rope. It will still result in climbing at acceleration of $a$.

 

[metastable]

Perhaps there’s 2 climbing phases, one when the body is lifted, and one when the arm is lifted, and each phase has a different mass & acceleration term?

 

[jbriggs444]

Perhaps there’s 2 climbing phases, one when the body is lifted, and one when the arm is lifted, and each phase has a different mass & acceleration term?

Or perhaps not. There is nothing that prevents the man from causing his center of mass to accelerate smoothly upward at the given rate. You are free to assume that the problem statement is correct.

 

[metastable]

But part of his body doesn’t accelerate at all during one phase (the hand).

 

[jbriggs444]

But part of his body doesn’t accelerate at all during one phase (the hand).

But that does not prevent the center of mass from accelerating smoothly upward at all times.

 

[A.T.]

But part of his body doesn’t accelerate at all during one phase (the hand).

Before you start your analysis, you first decide how to cut the system into bodies, and then stick to it. And yes, you can treat the hand and the rest of the person as separate bodies, just stay consistent about that.

 

[metastable]

If the climber is a small child with a very heavy hand or has a multiple kg wristwatch and we estimate the acceleration from the total body mass before the climb and force supplied, we could estimate an answer which is too low?

 

[jbriggs444]

If the climber is a small child with a very heavy hand or has a multiple kg wristwatch and we estimate the acceleration from the total body mass before the climb and force supplied, we could estimate an answer which is too low?

The hands are attached to the body. Their average velocity matches the average velocity of the body.

Why are you working so hard to make this problem harder than it is?

 

[metastable]

If we only look at the acceleration while the heavy hand grasps the rope, the hand won't accelerate, and no work is done on the hand, only to the rest of the body?

 

[jbriggs444]

If we only look at the acceleration while the heavy hand grasps the rope, the hand won't accelerate, and no work is done on the hand, only to the rest of the body?

While one hand is holding on, unmoving, the other hand is moving upward at double speed to get to the next grab point. It balances out.

 

[metastable]

Can we make the hand that's holding the rope as heavy as we want, without affecting the acceleration of the rest of the body with a given force (assuming only the hand that's holding the rope is heavy)?

 

[jbriggs444]

Can we make the hand that's holding the rope as heavy as we want, without affecting the acceleration of the rest of the body with a given force (assuming only the hand that's holding the rope is heavy)?

How can you climb hand over hand if you never move your hand?

 

[metastable]

That's why I thought it must be 2 phases with 2 acceleration and mass terms-- suppose the hands are different mass.

 

[jbriggs444]

That's why I thought it must be 2 phases with 2 acceleration and mass terms-- suppose the hands are different mass.

Suppose they are the same mass.

 

[metastable]

Then one phase lifts the body mass minus the hand mass, and the other phase lifts only the hand mass. The body could be simplified to a telescoping rod with 2 grappling claws. During each phase only one claw is grasping.

 

[jbriggs444]

Then one phase lifts the body mass minus the hand mass, and the other phase lifts only the hand mass. The body could be simplified to a telescoping rod with 2 grappling claws. During each phase only one claw is grasping.

There is one phase where the right hand is grasping and the left hand is rising. There is another phase where the left hand is grasping and the right hand is rising. Both phases are identical. There is no reason to worry about the distinction.

 

[.Scott]

A man tries to climb up a rope with acceleration, $a$. What does he actually do to climb up?

Why make things simple? Let's assume that this man will not only "try" to accelerate up the rope, but is taking on this project with real dedication.

So he will needs to apply the $F=m_m(g+a)$ that I specified earlier. As long as his velocity is up, he is "climbing the rope". But the acceleration doesn't have to be positive. He could start on a trampoline and then grab onto the rope and start climbing with not enough force to maintain his climb. He would reach a maximum height, and then drop back down to the trampoline. If I were that man, I would go with this option.

But most of us probably envisioned this man starting at the bottom of the rope with $V_0=0$. And by "climbing" we probably expect he will get at least a meter or two above the floor. We are also envisioning that the source of the mechanical energy is with the man - so something like a ski lift is out.

This arm movement stuff is also not going to cut it. I have tried climbing up a rope before. Doing it at all requires legs. Using arms to do it with constant velocity would be almost impossible. And doing it with constant acceleration is just not going to work.

He should use a space elevator - one where the "rope" stays in place and the gondola does the climbing. With nice computer controlled stepper motors, he will be able to control his acceleration precisely - and that nice constant acceleration is probably just what the elevator structure is best suited for. And of course, he will be able to maintain a constant acceleration for a much longer distance than most other methods - not just because of the length of the "rope", but because it will be extending into the vacuum of space where high velocities are better managed.

$F=(m_m+m_g)(g+a)$
$m_g$: mass of space gondola