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Is a hovering helicopter doing work?

  1. Sep 22, 2010 #1

    My background is ecology, but I am teaching a general intro to science class that covers many areas. I was covering the very basics of force and work today. I stated that energy is the ability to do work and gave work=mass*acceleration*distance. A student asked if a hovering helicopter is doing any work against the force of gravity because it (theoretically) is stationary in the vertical direction. I did not know how to answer this and said I would get back to him. My intuition that the helicopter is doing work conflicted with my snap evaluation of the implications of the work formula. Disregarding internal friction, it seems the helicopter blade is performing work by accelerating a mass of air downward. The air is also doing work against the blade. This is similar to a stationary fan moving air. Now, how does gravity come into this? Is it doing any work? Is the helicopter doing any work against gravity? Or is it simply 2 forces opposing each other? Thanks to anyone who can help me parse this.

  2. jcsd
  3. Sep 22, 2010 #2


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    Staff: Mentor

    You have the answer already in your head: Yes the helicopter does work against the air but there is no "output" since the the helicopter doesn't move. So in a sense the efficiency is zero.
  4. Sep 22, 2010 #3
    To add to what Russ wrote: Think of power as work per unit time. The helicopter's engine provides a certain amount of horsepower. Multiply that by the time on station, and you get the work done.

    In terms of efficiency in a hover, its measured as the ratio of ideal to actual power required, called the figure of merit. Typically FM = 0.7 is good.
    Last edited: Sep 22, 2010
  5. Sep 22, 2010 #4
    I would guess that there would be a general formula for calculating the work done by any propellor in any medium. For example, a certain boat propellor would move a certain mass of water with a certain amount of force per rpm, or something like that.

    As for the helicopter being stationary, I think that is a confusion factor (like the conveyer-belt with the plane taking off, if you're familiar with that conundrum). To get around the confusion, I think it would be best to consider what work the propellor would be doing if the helicopter was, say, flying horizontally with the same propellor rpm. Basically, the propellor generates a certain amount of thrust, and to do this they have to exert force over a distance.

    The fact that the helicopter is stationary only indicates that the center of gravity and angle of the propellor are situated in such a way as to perfectly oppose the thrust of the propellor.

    That's how I would look at it anyway - i.e. in terms of work done by the propellor blades as they exert force over a distance based on their speed and the air resistance they encounter.
  6. Sep 23, 2010 #5
    Thank you for the interesting replies. I think the confusion in my mind was/is considering a complex machine as a single thing. Ignoring the engine, the helicopter (minus propeller parts) is incapable of work if it just a weight hanging there. The propeller is obviously moving air and doing work and the air has an opposing force that balances the downward force of gravity if in a "stationary" hover. Zooming in to just the blade-air, it is obvious work is being done. The helicopter being stationary is superfluous because a weight (just matter) can't do work on its own (I would think, correct me please if I'm wrong). Now, consider a hovering helicopter and add a big weight to it but don't increase propeller speed. The helicopter would begin to descend but at a slower rate than usual. The propeller is then decelerating a mass over a distance and the confusion I had seems to evaporate in my mind. I think the question then comes to my mind that is in this situation of slow descent, is gravity doing work? I wouldn't think that a fundamental force as understood in Newtonian physics is capable of doing work. I'm probably just stating or mis-stating a basic principle here, so I'll do some more searching on this. Thanks again!!

  7. Sep 23, 2010 #6
    I think it would make sense to say the mass of the helicopter is doing work due to the force of gravity expressed by it. What's confusing me now is whether the amount of work the helicopter does in descending is the same as the amount it would do if it was descending with its engine stalled. Part of me thinks it would because the force of gravity is the same, as its its mass and distance to the ground. But it also seems like gravity would have to do more work to get the helicopter to the ground with its engine on. Certainly gravity has to act on the helicopter for a longer time to get it to the ground with the engine running. I wonder if this has to do with the difference between work and power, since I think power includes a time element whereas work is just force over a distance. Maybe someone else can explain this.
  8. Sep 23, 2010 #7
    Work is force times distance moved, as the helicopter is not moving it is doing no work. It is having to spend a lot of energy in doing so, but think about it this way if the helicopter had taken off, landed on a 100m high tower then shut down the engines you wouldn't ask if it was doing any work sitting on top of the tower.
  9. Sep 23, 2010 #8
    I've begun to wonder if one should account for the acceleration AGAINST gravity when calculating work done when hovering? If no counter force is present -- rotors spinning or muscles holding an arm up -- the object would be falling at 32 ft/s^2, and thus moving and doing work. Or is that the wrong way to think about it?
  10. Sep 23, 2010 #9


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    Nope. If the helicopter is not moving, it isn't doing any work (on itself). It's exerting a lot of force, but the distance travelled is zero, so the work is also zero. This is somewhat counterintuitive at first, but that's simply because of the way in which the helicopter is generating force. The blades are doing work on the air, and the helicopter as a whole (as a result) is doing work on the air, but there isn't any work being done on the helicopter itself. The same would be true if the helicopter were just sitting on the ground - there would be force exerted on the skids by the ground, but since the helicopter isn't moving, no work is done.
  11. Sep 23, 2010 #10
    ah, so one would measure "work" relative to something? So rotors vs air-speed might produce a work value but not rotors vs ground? Then the analogy won't extend to doing work by holding ones arm our horizontally.... And here I thought basic physics was intuitive...
  12. Sep 23, 2010 #11
    Helicopters don't change their rotor speed in hover, they change their collective blade pitch angle. Helicopter rotors operate differently than an aircraft propeller does.
    Last edited: Sep 23, 2010
  13. Sep 23, 2010 #12
    In a hover, gravity does no work on the helicopter. Work is the product of force times distance.

    Work is the product of force times distance.

    That's right, because: Work is the product of force times distance.

    No: Work is the product of force times distance.

    No: Work is the product of force times distance. Time is not found in this definition.

    Yes, power is work per unit time.
  14. Sep 23, 2010 #13
    No, airspeed is zero in a hover, by definition. The power needed by the rotors to accelerate a mass of air downward can calculated via momentum theory or blade element analysis. The work done by gravity, however, is still zero.
  15. Sep 23, 2010 #14
    The question needs to be defined rather carefully:

    Well the helicopter IS doing work against friction of internal parts and air flow, and IS doing additional work moving air. Just like a pump does work moving water. And work is also being done in heating the fuel during combustion and in frictional heat losses.

    But the strict definition of work is force times distance so the helocipoter itself, being stationary, is NOT doing a work in that sense. It is burning fuel to maintain a fixed hover, as you burn food calories sitting motionless in a chair, but neither of those involves work because while forces are clearly involved there is no motion (no distance). I can push as hard as I wish against a wall, but if it remains motionless, no physical work is done...except maybe a tiny bit of thermodynamic warming....

    A discussion of mechanical work is here:http://en.wikipedia.org/wiki/Work_(physics [Broken])

    and thermodynamic work here: http://en.wikipedia.org/wiki/Work_(thermodynamics [Broken])
    Last edited by a moderator: May 4, 2017
  16. Sep 23, 2010 #15

    Filip Larsen

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    Gold Member

    When talking about work and power and such, it most often helps a lot if you define system boundaries and then invoke conservation laws for that system.

    For instance, you could define the system to be a narrow "box" enclosing the helicopter and the air immediately around it, meaning you would have to include the energy content of the air entering and leaving the box in order to find out what the work on (or by) the system is. Assuming a steady state situation, the total power of the airflow out of the box would then equal the power added by gravity and engine.

    You could also define the system to be a huge box around the helicopter such that you can assume no air passes in or out of the box (like if it was a solid box). In this case the total work on the system is zero, which then means that all the energy released by the engine goes into increasing the total energy of the air inside the box (which then can't be a steady state flow), and that gravity does very close to zero net work on the mass in the box.
  17. Sep 23, 2010 #16
    However the air is not stationary.
    A helicopter in hover is exactly equivalent to a fan at the end of a tube, it is compressing the air underneath it.
    The helicopter is able to stay in the air because of the extra positive pressure of the air underneath it times area of it's rotor blades is equal to the weight of the aircraft.

    Because the system is rather lossy (there isn't a tube to hold the cushion of air) a helicopter has to do quite a lot of work.
  18. Sep 23, 2010 #17


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    Staff: Mentor

    Yep. The helicopter is doing lots of work on the air. So, for the title of this post, "Is a hovering helicopter doing work?", then I'd have to say yes. It is.
  19. Sep 24, 2010 #18
    Excellent. Just like entropy. Thanks!
  20. Sep 24, 2010 #19
    If your student is asking about whether the hovering helicopter is doing work against gravity, well, no. Imagine there is no air, only gravity, the helicopter would never be able to hover (it won't fall if its horizontal velocity is about (gR)^(1/2) which is not hovering ) So, is a hovering helicopter doing work against the air ? Yes. And why bother with the mechanism of a helicopter in such question =_=
    Last edited: Sep 24, 2010
  21. Sep 24, 2010 #20
    Uhmm... and if you have the opportunity to stand beneath or at least near a running helicopter, you will note that air, and a lot of it, is both moving and forcing against things.
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