# Work done lifting a box

1. Apr 25, 2014

### katrema

I need some help understanding work.

If a man lifts a box up to a shelf and a woman lifts the same box up to the same shelf in half of the time, who is doing more work?

I understand that they're moving it the same distance but isn't the woman using more force because she is getting it to the shelf faster?

2. Apr 25, 2014

### UltrafastPED

The mechanical work done is the same; but people are not ideal machines, so we will always expend more energy than the amount of work that is done.

Assuming that the force is applied in the direction of motion, you can calculate the mechanical work as
Work done = Force x Distance.

If two different machines are used to lift a box, it should require the same amount of work to get the box to the same height - the box will have the same potential energy due to gravity if it is moved to the same height:
Potential = m*g*h.

So note that in your situation each person (or machine) is using a varying amount of force at different stages of the lift - it has to be less at the top of the lift, or the box would continue to go up!

In this case you have to add the forces and the distances in small steps (an integral is the idealization of infinitely small steps).

The work will be the same, but the power (rate of use of energy, energy per unit of time) is different.

3. Apr 25, 2014

### Simon Bridge

Welcome to PF;
The work done on the box is the same as it's change in energy.
In this case, it's the gravitational potential energy that changes so W=mgΔy

The man has just used more power to do the same work faster.

Notice that W=Fd means that F=mg ... which would move the box at a constant speed.
So what about the bigger acceleration?

Applying more force than F=mg means that the box also ended up with a lot of kinetic energy at the end of it's journey, the man may have put some effort to decelerate the box to put it on the shelf (or just let gravity slow it). This does negative work on the box to cancel out the kinetic energy.

Since neither man nor woman are perfectly efficient, the work they did is greater than the work done on the box.

What you left out of your description is the details of the forces used to get the box on the shelf.

4. Apr 25, 2014

### 256bits

Just so you do not get bogged down on whether the man or the woman does more work, you question should be:
If a man lifts a box up to a shelf and a woman lifts the same box up to the same shelf in half of the time, who is doing more work ON THE BOX?

The work ON THE BOX is the same in both cases.

Taken literally, whether the man or the woman is doing more work actually cannot be said. The woman could be shorter than the man and cannot reach the shelf but has to use a stool to step onto to place the box. Here she would have to raise her whole body upwards from the floor and could end up doing more work than the man who might have to raise only his torso.

Of coarse, what you were asking here was completely understood, but in another situation if the explanation is not explicit, there could end up some disagreement as to what is being discussed.

5. Apr 25, 2014

### dauto

Moving faster doesn't mean more force is used. Newton's 2nd law says F=ma. The speed doesn't show up in that formula does it? In that case both the man and the woman are producing just enough force to cancel gravity producing zero acceleration and constant speed. The speed they use is different but they are both constant.

6. Apr 25, 2014

### Staff: Mentor

I disagree. When lifting a box in a short time, the acceleration is not negligible.

7. Apr 25, 2014

### Simon Bridge

Care needs to be taken about what sort of assumptions are made. The problem statement is fairly interpreted to include: the box starts and ends it's journey at rest. The above (quoted passage) is only true if the box starts and ends it's journey in uniform motion.

It may be that a constant force was applied through most of the motion, and, therefore, most of the motion was uniform ... but there must have been some time during which the acceleration was not zero, or the box never leaves the ground.

The person who lifts the box to the shelf in the shortest time has done the most work - as OP noticed. But the box received the same amount of gravitational potential energy for everyone as long as the shelf and floor are always the same distance apart.

It's one of those "devil in the details" things.

For instance, we can do all the work at the start of the journey by giving the box a kick.

8. Apr 26, 2014

### UltrafastPED

It seems to me that if the box is simply hoisted to the level of the shelf, then integrated force x distance along the vertical lift will result in the same work done - if the original force is a bit more at the beginning, it will be correspondingly less as you approach the top.

The faster hoist requires more average power, but the total energy budget should be the same for an ideal machine. People of course are not machines, and it is not clear at all (to me) which person would expend more total energy in lifting a box.

Am I missing something in the details here?

9. Apr 26, 2014

### Simon Bridge

That would give you the work done by that force. Which force did you have in mind?

The work done on the box is the change in energy of the box.

The person who does the most work is the one who expends the most energy in the task.
The method of doing the work is important - if the person threw the box to the shelf, then they do work giving the box it's initial kinetic energy. The box with the most kinetic energy arrives at the shelf soonest - and the shelf+wall will have to do some work to stop the box.
Thus, the person with the fastest time does the most work - even if the person were an ideal machine.

But if the woman were shorter, and she climbs a ladder to carry the box up, then she has to do some extra work lifting herself and the box up the ladder, which takes a while, while the man just lifts the box... in that case, the slower person does more work.

The work done on the box, against gravity, is the same for each case and does not depend on how the work was done because gravity is a conservative force.

Like I said: it's a devil in the details problem.

10. Apr 26, 2014

### gmax137

Simon, at the risk of seeming a bit PC, I think you might be confusing the readers with your responses about the man being faster. Please note the OP's scenario:

Yes the devil is in the details...

11. Apr 26, 2014

### Simon Bridge

@gmax - drat. Well spotted.
I'm not the only one got that mixed up either. Too late to edit.
Hopefully anyone who read this far managed to realize the mixup.

JIC:
Depends on how the box gets to the shelf.

The work done on the box is the change in energy of the box - but not all the work done by the two people need go into changing the energy of the box.

The man may have been very short and needed to climb a stepladder to put the box on the shelf, while the woman managed it in one go. Therefore the man had to do work to raise himself up the ladder as well as work to raise the box. So the man may end up doing more work, even though he was slower.

If the box was kicked onto the shelf, the it will arrive at the shelf with some extra kinetic energy - since the woman got it to the shelf faster, then her box arrives with more kinetic energy, she she had to have done more work.

When formulating the work by force times distance, you have to be careful about which force you are using - that is the force that does the work. i.e. for both scenarios, the force of gravity does a total W=-mgh work on the box.

That better?