Conceptual Question on Work and Negative Work

[jon4444]

I'm confused by classic description of work and negative work. If someone pulls (slowly) on a rope to lift a bucket in a well, I understand that the person is doing work on the bucket and gravity is doing negative work on the bucket. So, the net work is zero.

But if the net work is zero, where is the energy coming from that increases the bucket's potential energy?

 

[arpon]

At first, consider the following example:
Suppose, you have thrown a ball(mass=$m$) from ground to the sky with velocity $v$;
So the ball has a kinetic energy = $\frac{1}{2}mv^2$ ;
After that, the net force on the ball is just the gravitational force which is opposite to the displacement;
So, the work is now negative, which means, the kinetic energy is decreasing and the potential energy is increasing.
Negative work means decreasing of kinetic energy and increasing of potential energy.
Just look at the mathmetical calculation:
$W = Fs$, where $W$ is work, $F$ is force, $s$ is displacement;
$=mas$, $a$ is accelaration,as $F=ma$
$=\frac{1}{2}m*2as =\frac{1}{2}m*(v^2 - u^2)$, $u$ is the initial velocity; and we know, $v^2 = u^2 + 2as$
$=\frac{1}{2}m*v^2 - \frac{1}{2}m*u^2$
=final kinetic energy - initial kinetic energy
=change of kinetic energy
so, if the change of kinetic energy is negative, the potential energy will increase to conserve energy.
When the ball reaches the highest level, the velocity is then 0 ; and the potential energy is $mgh$, $h$ is the height;
After that the ball will fall downward in the direction of gravitational force, so the work is positive and the kinetic energy will increase.

Now come to your case.
If you suppose, the rope is frictionless, then the case is:
At first, you have to apply some force upward on the rope (which is greater then m*g);
as, the force is greater than the gravitational force, the bucket will have net force upward;
the displacement is in the direction of net force, so there will be a positive work;
the velocity of the bucket will be increasing(upward) for a little bit of time, the kinetic energy will also increase;
and then you will just pull the rope with only the force equal to gravitational force; and the net force is equal to 0; so 0 work; there is no accelaration; no increase or decrease of velocity; so, no change of kinetic energy;
at last you don't apply any force on the rope (for a little bit of time), and there is a net force downward (equal to gravitational force) opposite to the direction of displacement; negative work; decrease of velocity and kinetic energy; increase of potetial energy;
when the bucket stops (velocity = 0), then you have to exert a force equal to gravitational force upward so that the net force is 0;

Got it?

 

[jon4444]

It's helpful to add the kinetic energy into the situation with the bucket, but I still don't understand how the net work is zero if the system of the bucket and the Earth starts with a certain potential energy and then ends with increased potential energy. (If you just look at the initial and final state, there's no kinetic energy, so the total mechanical energy is all potential energy and it has increased.) Why wasn't positive work needed to increase the potential energy?

 

[arpon]

Why wasn't positive work needed to increase the potential energy?

At first, you have to apply some force upward on the rope (which is greater then m*g);
as, the force is greater than the gravitational force, the bucket will have net force upward;
the displacement is in the direction of net force, so there will be a positive work;

As you can see, at first there is a positive work;

 

[jon4444]

As you can see, at first there is a positive work;

But then why do AP Physics books say the net work is zero (positive work pulling up on rope completed balanced out by negative work of gravity)?
(I suppose the logic here is that your initial force has to be greater than mg to get bucket moving, then constant velocity going up means net force is zero, then when bucket comes to stop, your force is momentarily less than gravity, so everything theoretically balances out, but still there's an increase in potential energy.)

 

[jon4444]

Is the answer that if you're including gravitation pull of Earth in system, then there really is no "external" potential energy, because if the bucket were to translate the potential energy into kinetic energy by falling, that would be gravity doing positive work on the bucket.

 

[arpon]

I apologise as my last post (#4) was not the appropriate explanation.

If I haven't mistaken, your question is that, initially there is a positive work, and finally a negative work, which are equal to each other. So the mechanical energy should be constant.

Actually, it is constant.
If you go through the problem deeply, you notice that, initially there is a positive work, which means increase of kinetic energy. But from where the kinetic energy come? It comes from the energy of your 'muscles'.
Finally, the negative work converts the kinetic energy into potential energy.

So you can seeIt is just the conversion of energy.
http://attachments/untitled-jpg.74752/?embedded=1&temp_hash=295cb5f590b594f99e49802f171746ef
Please feel free, if you have further questions.

 

[Dale]

In this type of problem you have to be careful about how you define your system. Usually the confusion comes in when you're implicitly defining the system in different ways in different parts of the problem.

One way to define the system is the say the only the bucket is the system. In that case, the gravitational potential energy is not part of the system. The gravitational potential energy belongs to the gravitational field of the Earth, not the bucket. With that definition the rope does positive work on the system, and gravity does negative work on the system, with the net work being 0, and a net transfer of energy from the rope to the Earth and its gravitational field.

Another way to define the system is to say that the bucket and the Earth together are the system. In that case, the gravitational potential energy is part of the system. With that definition the rope does positive work on the system, and gravity is an internal force doing no work on the system. Now, there is a net transfer of energy to the system.

The confusion comes when you consider the bucket to be the system, but then consider the gravitational potential energy to be part of the system also.

 

[jon4444]

In this type of problem you have to be careful about how you define your system. Usually the confusion comes in when you're implicitly defining the system in different ways in different parts of the problem..

So, if you define the system as bucket and earth, again, how can total work be zero and yet there be an increase in potential energy?

 

[Dale]

So, if you define the system as bucket and earth, again, how can total work be zero and yet there be an increase in potential energy?

In that case the total work is not zero, it is positive. The internal force of gravity does not do work on the system, only the external force from the rope.

 

[jon4444]

In that case the total work is not zero, it is positive. The internal force of gravity does not do work on the system, only the external force from the rope.

OK, I think that's the breakthrough I was looking for.

Say you define the system as including both Earth and guy pulling rope as well, in which case all work is internal and there's no potential energy since if the bucket were to fall that would involve it pulling on rope?

(And so I don't appear stupid, I think you added a paragraph after I wrote my follow-up question, which pretty much clarified it for me...)

 

[Dale]

I am not sure that I would say that they are poorly thought out. You can make things extremely exact and correct, but then you tend to make it more difficult to learn. It is generally better to introduce things gradually rather than hitting students with all of the potentially messy details at once.

I would like for textbooks to do a better job of pointing out to students about how flexible "the system" is, but how important it is to actually be clear about what you are defining as "the system" in any problem. Here you could define the system in any of the following ways (plus an infinite number of other ways):

Bucket only
Bucket + Earth (including gravitational field)
Bucket + Earth + Rope
Bucket + Rope
Bucket + just part of rope (you don't have to include an entire object in a system)
Bucket + Rope + Person
Etc.

You can get a correct analysis with every system listed above, but different definitions of the system will change different forces from being internal to being external.

 

[Dale]

And so I don't appear stupid, I think you added a paragraph after I wrote my follow-up question, which pretty much clarified it for me...

Yes, sorry about that. I try not to edit after I get a reply, but sometimes I still wind up editing at the same time someone is replying.

 

[jon4444]

No problem. And if you wouldn't mind addressing the final situation I added--- if you define the system as everything (earth, rope, guy pulling rope, bucket), then I assume total work is zero (no external forces), but there is an increase in gravitation potential energy--is that offset by the decrease in "rope potential energy" (or however you might describe fact that if bucket drops, it will pull on rope and against guy attached to rope)

 

[jbriggs444]

Some of the chemical energy in the guy's muscles will have been converted into gravitational potential energy. That balances the energy books. Some additional chemical energy will have been wasted in other ways, of course.

If you focus in and consider the guy as a system, his two external interfaces (hands pulling on the rope and feet pushing on the earth) are injecting net energy (doing positive work) into the outside world.

Similarly, if you focus in and consider "gravity" as a system, its external interfaces (pulling up on the Earth and pulling down on the bucket) are subtracting net energy (doing negative work) from the outside world.

 

[Dale]

there is an increase in gravitation potential energy--is that offset by the decrease in "rope potential energy"

As jbriggs444 mentioned, the increase in gravitational PE is offset by the decrease in chemical energy (neglecting heat).

 

[jon4444]

As jbriggs444 mentioned, the increase in gravitational PE is offset by the decrease in chemical energy (neglecting heat).

Thanks. Then I'm assuming that if you slowly lower the bucket, and are still looking at a system of everything, classical energy conservation becomes a problem because the chemical energy in your muscles isn't truly a conservative force. So the PE of the bucket might just get transferred into heat loss of your muscles as they eccentrically lower it...

 

[Dale]

Heat is part of classical mechanics so there is no problem, just one more place where energy can go.