Where does the energy go when lifting a rope?

[Order]

Homework Statement

A uniform rope of mass $$\lambda$$ per unit lengt is coiled on a smooth horizontal table. One end is pulled straight up with constant speed v₀.

a. Find the force exerted on the end of the rope as a function of height y.

b. compare the power delivered to the rope with the rate of change of the rope's total mechanical energy.

Homework Equations

$$F=\frac{dp}{dt}$$

$$P=Fv$$

$$\frac{dE}{dt}$$

The Attempt at a Solution

a. I have no problem in finding the solution. I use $$p(t)=v_{0}t \lambda v_{0}$$ where $$y=v_{0}t$$and $$p(t+\Delta t)=v_{0}(t+\Delta t) \lambda v_{0}$$ to obtain $$\frac{dp}{dt}=\lambda v_{0}^{2}.$$ And therefore $$F=\frac{dp}{dt}+F_{g}=\lambda v_{0}^{2}+\lambda y g.$$ Here the first term is independent of height which is surprising to me, but never mind.

b. I use $$P=Fv=\lambda v_{0}^{3}+\lambda v_{0}yg.$$ Now to find the energy change I use $$E(t)=v_{0}t \lambda \frac{v_{0}^{2}}{2}+\lambda y g \frac{y}{2}=v_{0}t \lambda \frac{v_{0}^{2}}{2}+\lambda g v_{0}^{2}\frac{t^{2}}{2}$$ $$E(t+\Delta t)=v_{0}(t+\Delta t) \lambda \frac{v_{0}^{2}}{2}+\lambda g v_{0}^{2}\frac{(t+\Delta t)^{2}}{2}$$ and therefore $$\frac{dE}{dt}=\lambda \frac{v_{0}^{3}}{2}+\lambda v_{0}yg.$$ You can see here that gravitational energy is conserved whereas the energy change because of momentum is not. There are no collisions, so where does the energy go?

 

[Doc Al]

You can see here that gravitational energy is conserved whereas the energy change because of momentum is not. There are no collisions, so where does the energy go?

In a sense there is a 'collision'. Realize that the lifted rope segment is not slowly increased in speed, but jerked from zero speed to the final speed v. The extra energy used ends up as thermal energy.

 

[Quinzio]

I think the problem here is very subtle and require some more thoughts.
Doc Al is good in understanding that the point where the rope starts lifting is crucial for a proper comprehension of the problem.
Although it's not correct to state that some energy is lost into heat.
Of course in a real world this would happen in some quantity, but an ideal rope (perfectly flexible) would not lose energy.

The point is to divide the problem into 2 different situations:
a. the part of the rope which is on the floor is NOT sliding
b. the part of the rope which is on the floor is sliding

this changes a lot as we will see.

case a.
The correct way to represent the problem is in the next figure:

[PLAIN]http://img864.imageshack.us/img864/8817/ropea.jpg

If the part of the rope which lies on the floor is not sliding, then while we are lifting the rope, the point E goes in point D, and the corner made by the rope goes from D to B.

I leave to the Original Poster to calculate the total kinetic energy rate of change in this case (this is the homework section, so no complete solution at first). The result is the same than the power given to the cord:
$P=Fv=\lambda v_{0}^{3}+\lambda v_{0}yg$
So, no energy is lost.

The case b is trivial, since all the rope is moving, so that the vertical part is rising and the horizontal part is sliding on the floor (no friction involved).
When the head of the rope rises by 1 mm, the tail of the rope on the floor moves by 1 mm.
A goes to B while C moves to D.
[PLAIN]http://img7.imageshack.us/img7/6452/rope2e.jpg

NB: The problem states that the rope is "coiled", reminding of a real world situation (e.g. on boats). Even in this situation the problem can be thought as case "a", with some additional remarks.

 

[Doc Al]

Quinzio:

I believe you are solving a different problem than that of the OP. As I understood it, the rope is coiled on the table (not laid out in a line) while one end is pulled straight up (not peeled off at an angle) at constant speed. I agree with the OP's solution. And there's definitely a loss of energy to heat as the rope segments are jerked to full speed.

I do agree that it is a subtle problem.

 

[Quinzio]

Ok, but in all situation the part of rope on the ground must fall in one of the case I mentioned before, or in a mix of the two.
It doesn't exist a rope which is created "out of nothing" from the ground as long as the top is pulled upwards.

The most realistic case is a rope coiled in a perfect circle. Of course the rope is ideal, so it's perfectly flexible and has a diameter zero (or negligible). The rope leaves the ground in the center of the circle, and a piece of rope unfolds from the circle to the center as long as the rope is pulled upwards.
In this case it seems that the rope on the ground has always the same energy, but a closer look is necessary. Say the piece of rope from circle to center "turn" with an apparent angular speed omega. Every point of the rope, as it approaches the center, tries to retain it's linear speed, so it increases it's omega. In order to maintain omega constant, some energy must be dissipated.

In any case, no energy goes in heat. An ideal rope cannot dissipate energy, it's like an ideal chain without friction.
Even for a real rope, I think that it sounds a bit odd to say that flexing it will generate some heat.

 

[Order]

The most realistic case is a rope coiled in a perfect circle. Of course the rope is ideal, so it's perfectly flexible and has a diameter zero (or negligible). The rope leaves the ground in the center of the circle, and a piece of rope unfolds from the circle to the center as long as the rope is pulled upwards.
In this case it seems that the rope on the ground has always the same energy, but a closer look is necessary. Say the piece of rope from circle to center "turn" with an apparent angular speed omega. Every point of the rope, as it approaches the center, tries to retain it's linear speed, so it increases it's omega. In order to maintain omega constant, some energy must be dissipated.

I am more confused than ever. I try to bring some order to it, but can't, although that's my name.

I tried to solve the problem with a coiled rope and had a center of mass speed of the rope of v₀/2 in the omega direction, but I did not take into account what you said here Quinzio, that the "rope tries to retain its linear speed". However the result is still that half the "first term"-energy is lost. And now you say it is even more. (I won't bore you with my calculations.)

I agree that your model gives twice the value of dE/dt because you add speeds in both the x- and y-direction, but then dP/dt*v is also twice as large because of the very same reason, so even in your model half of the energy is mysteriously lost. Isn't that so?

 

[Order]

Quinzio:

I believe you are solving a different problem than that of the OP. As I understood it, the rope is coiled on the table (not laid out in a line) while one end is pulled straight up (not peeled off at an angle) at constant speed. I agree with the OP's solution. And there's definitely a loss of energy to heat as the rope segments are jerked to full speed.

I do agree that it is a subtle problem.

I do not quite understand how a jerk would lead to loss in energy. In a real situation the rope will not be fully flexxible but a little stiff and therefore will be slowly accelerated. But this does not change dE/dt I think because the acceleration parts are the same in both E(t) and E(t+dt) and therefore cancel. Energy is lost without the jerk.

 

[Antiphon]

The answer is not in the jerk. You could lift the rope over a billion years and the energy is still finite.

It goes into increasing the gravitational field of the planet/rope system. If you don't want to invoke GR then it simply moves the energy to gravitational potential energy. Think of it as winding a gravitational spring.

 

[Quinzio]

I do not quite understand how a jerk would lead to loss in energy. In a real situation the rope will not be fully flexxible but a little stiff and therefore will be slowly accelerated. But this does not change dE/dt I think because the acceleration parts are the same in both E(t) and E(t+dt) and therefore cancel. Energy is lost without the jerk.

No energy is lost and no heat is produced with an ideal rope.
The problem is that there no real way to lift a rope from the ground and get this two conditions:
a- the rope on the ground doesn't move
b- the rope off the ground is perfectly vertical.

One of these two conditions will always be broken (likely together), and so the movement of the rope will be quite complex and nearly impossible to determine analytically.

The only real way to achieve conditions a and b is two move the rope very very slowly. In this way the cube of the speed is indeed negligible and the conditions a an b are satisfied with a small error.

 

[Order]

I seem to agree with Doc Al, although I don't understand really how heat can be generated by a jerk. The reason is that if the rope is slowly accelerated the power P and energy change dE/dt are identical. Let me show this below without including the gravity term.

First let the mass of the rope as a function of y be so that $$m(0)=\infty$$ and $$m(y)=\lambda y.$$ Also let the velocity of the rope as a function of y be so that v(0)=0 and v(y)=v₀. For example I can choose $$m(t)=\lambda \frac{y_{0}^{2}}{y}$$ and $$v(t)=v_{0}\frac{y}{y_{0}}.$$ According to the calculations in my original post this will still lead to a force $$\frac{dp}{dt}=\lambda v_{0}^{2}$$ but the power will be $$P=v\frac{dp}{dt}=\lambda v_{0}^{3}\frac{y}{y_{0}}.$$ I am interested in the mean power which is $$P_{mean}=\lambda \frac{v_{0}^{3}}{2}.$$ The energy will be $$E(t)=\lambda \frac{y_{0}^{2}}{y}v_{0}^{2}\frac{y^{2}}{y_{0}^{2}}\frac{1}{2}=\lambda v_{0}^{3} \frac{t}{2}$$ and then of course the energy change will be $$\frac{dE}{dt}=\lambda \frac{v_{0}^{3}}{2}.$$

I don't know if this model looks strange, but I think it is perfectly realistic.

 

[Order]

The answer is not in the jerk. You could lift the rope over a billion years and the energy is still finite.

It goes into increasing the gravitational field of the planet/rope system. If you don't want to invoke GR then it simply moves the energy to gravitational potential energy. Think of it as winding a gravitational spring.

But what if I lift the rope in a zero gravitational field, for example in orbit around the earth?

 

[Antiphon]

It would take almost not work to lift it then. You'd just have to start it moving and it would drift the rest of the way.

 

[Order]

Ok, I think I need to say just a few more things. There is far too much to say really, but it will be best not to say too much. I am still confused about this problem and how power is defined and executed. I wish there was a physics hero that could explain to me but I guess there is no one. This said, I would nevertheless like to thank Quinzio for his detailed remarks.

Now there is one more thing I would like to ask before digging deeper into other problems. (I don't like to have several posts at once in the forum.) Anyway, the question is, which one of my two models is most realistic? I have one where there is a jerk and one that avoids jerk. Is there any jerk in reality, or is there not?