Minimum force to keep a mass floating in the air

[guest1]

This is bugging me and maybe someon can help me find the obvious mistake:

I want to calculate the minimum power needed to keep a mass of 1 kg floating in mid air.

The naive approach is this:

The mass falls 5 meters in one second so to get to the original position I have to invest an energy of 50J (m*g*h with g = 10 m/s^2) per second

So for the one second case I get a power need of 50W.

If I do the calculation for 2 seconds I need 100W.

Actually what I would have to do is get the limit for t->0 ...but of course that gives me 0W (which is not very helpful since stuff like airplanes and helicopters do require constant lift/power to keep falling out of the sky)

 

[mgb_phys]

No power is required to keep something in the air.
A shelf manages to do it for years without any power source.

Energy is force * distance, (and power is just energy/time) if you don't move anything against a force you don't need any energy.

The reason heavier than air craft need power is that they are pushing an aerofoil (ie. wing) against the force of aerodynamic drag (the rate of pushing is power) - even helicopters hovering are pushing a rotor blade around.

 

[russ_watters]

Use a table: power = 0

 

[guest1]

Zero is what I get, too - but that's not a real useful answer.

The question came about like this: Let's say I want to build some sort of flying contraption that is able to lift the mass of 1kg into the air. What would be the lowest power energy source I would need for this (in the absence of some sort of passive buoyancy aid) .

If you take a 1kg mass and hold it up in the air you are expending energy/power (mathematically it's zero, but you really are using calories to accomplish the feat otherwise you could do it indefinitely/effortlessly. You need to counteract the force downwards constantly. This should be similar to the drag issue, shouldn't it? (there you are constantly counteracting a force also))

 

[Doc Al]

The question came about like this: Let's say I want to build some sort of flying contraption that is able to lift the mass of 1kg into the air. What would be the lowest power energy source I would need for this.

How far do you want to lift it? How fast? As has been pointed out, no work is required to support the mass once you've lifted it into position. (And put it on the table.)

If you take a 1kg mass and hold it up in the air you are expending energy/power (see if you can do it indefinitely. You need to counteract the force downwards. This should be similar to the drag issue, shouldn't it? (there you are constantly counteracting a force also)

Sure it will require energy for you to support it with an outstretched hand. But that's because you chose a biological system to support the mass. All of that energy expended to create tension in your arm to support the mass is wasted. Use a table.

 

[guest1]

How far do you want to lift it? How fast? As has been pointed out, no work is required to support the mass once you've lifted it into position. (And put it on the table.)

I know that one could lift the mass (theoretically) infinitely slowly using a 0.0001W source. But in reality that just doesn't work (if you attach it to a string and pull it upwards with such a power source it would lift off the floor but I'm talking about flying contraptions here)

You can rig a system that will generate a lifting force to the mass (e.g. a very slowly turning helicopter rotor with a 0.0001W motor) and it will never get off the ground even though you are constantly enacting a force upwards.

Obviously this is because the force needs to be greater than the gravitational attraction.
The point is: You need to generate that force somehow so some sort of energy is involved (and hence some sort of power) otherwise we could scale down flying machines indefinitely (which seems absurd).

 

[Doc Al]

I know that one could lift the mass (theoretically) infinitely slowly using a 0.0001W source. But in reality that just doesn't work (if you attach it to a string and pull it upwards with such a power source it would lift off the floor but I'm talking about flying contraptions here)

This is the first time you've mentioned 'flying contraptions'.

You can rig a system that will generate a lifting force to the mass (e.g. a very slowly turning helicopter rotor with a 0.0001W motor) and it will never get off the ground even though you are constantly enacting a force upwards.

Obviously this is because the force needs to be greater than the gravitational attraction.
The point is: You need to generate that force somehow so some sort of energy is involved (and hence some sort of power) otherwise we could scale down flying machines indefinitely (which seems absurd).

You want to know what kind of efficiencies are available for helicopter-like systems? What is the minimum power required to generate a certain thrust?

 

[stewartcs]

I know that one could lift the mass (theoretically) infinitely slowly using a 0.0001W source. But in reality that just doesn't work (if you attach it to a string and pull it upwards with such a power source it would lift off the floor but I'm talking about flying contraptions here)

You can rig a system that will generate a lifting force to the mass (e.g. a very slowly turning helicopter rotor with a 0.0001W motor) and it will never get off the ground even though you are constantly enacting a force upwards.

Obviously this is because the force needs to be greater than the gravitational attraction.
The point is: You need to generate that force somehow so some sort of energy is involved (and hence some sort of power) otherwise we could scale down flying machines indefinitely (which seems absurd).

The minimum force required to keep an object "floating" on Earth is equal to the object's weight. This can be accomplished with buoyancy or some other means.

One can build many different machines to achieve the "other means" such as helicopters. However, it becomes a matter of which has the best efficiency. The one with the best efficiency will use the least power to keep the object floating.

CS

 

[guest1]

This is the first time you've mentioned 'flying contraptions'.
You want to know what kind of efficiencies are available for helicopter-like systems? What is the minimum power required to generate a certain thrust?

Sort of. I was simply wondering whether there is a lower limit for the power needed to generate the kind of thrust which would allow a certain mass to remain stationary in the air (or lift off infinitely slowly).
I'm specifically talking about thrust here not any sort of buoyancy or lift generated by wings and airspeed. Just plain thrust downwards from a standing start.

Whether this is by helicopter or other means (e.g. particle acceleration via electromagnetic thrusters or whatnot) is immaterial.

This is the first time you've mentioned 'flying contraptions'.

I realize now that the term 'floating' was a bit ambiguous in the OP, sorry

 

[russ_watters]

There is no theoretical lower limit to the power. For a helicopter, the larger the rotor, the lower the power.

 

[mgb_phys]

Sort of. I was simply wondering whether there is a lower limit for the power needed to generate the kind of thrust which would allow a certain mass to remain stationary in the air (or lift off infinitely slowly).

Not really, you just have to fly slower with a larger wing.
eg http://en.wikipedia.org/wiki/Solar_Impulse_Project

 

[guest1]

The point was: pure vertical thrust. no wings. no horizontal component. no balloon.

There has to be some way to calculate the minimum impulse I need to impart on air (or the amount and speed of reactive mass) in order to keep station?

 

[russ_watters]

Again, there is no minimum.

 

[guest1]

Somehow I can't really believe that. If you make the rotorblades on a helicopter wider then you can turn it slower but the amount of air you need to shove downwards remains the same. You need to continually shove some air downwards or your object will fall. You can't reduce that to zero. Moving air downward means imparting an impulse to it (i.e. accelerating it) which means imparting energy to it which means that there is some power involved.

Likewise with something that uses a reactive mass (rockets). In order to at least keep their height you need to continually expel a certain amount of mass downwards. Gravity wants to impart in impulse downwards on your object so you need to counteract that in some way.

 

[TurtleMeister]

I think guest1 wants to know how much power (watts or horsepower) is required to accelerate a 1kg mass at 9.8 meters per second per second.

 

[Dale]

Somehow I can't really believe that. If you make the rotorblades on a helicopter wider then you can turn it slower but the amount of air you need to shove downwards remains the same. You need to continually shove some air downwards or your object will fall. You can't reduce that to zero. Moving air downward means imparting an impulse to it (i.e. accelerating it) which means imparting energy to it which means that there is some power involved.

Likewise with something that uses a reactive mass (rockets). In order to at least keep their height you need to continually expel a certain amount of mass downwards. Gravity wants to impart in impulse downwards on your object so you need to counteract that in some way.

You need to generate a force equal to the weight, which means that you need to accelerate some mass of air downwards. The question is how much air mass you accelerate. The more air you accelerate the less you have to accelerate it and therefore the less power is needed. There is no theoretical limit to this. The larger mass of air you push down the less you need to accelerate it and therefore the less power you require.

Essentially, your error is in the assumption that the amount of air you need to shove downwards remains the same. This is not correct. You can shove more or less air downwards depending on how much you accelerate the air. A small-mass high-velocity air movement will be much more energetic than a large-mass low-velocity air movement.

 

[TurtleMeister]

I'm sure that it gets complicated when considering real world application such as helicopters. Just google "power required to hover". However, the question that I am considering is: What is the power required to accelerate a 1kg mass at 9.8 ms^-2? (disregarding exactly how we're going to do it)

Here is my attempt. I'm not sure that it's correct so please check my work.

Converting weight to force:
acceleration = 9.8ms^-2
mass = 1kg
Force = mass * acceleration = 9.8 Newtons

Work done in moving 9.8 meters:
Work = Force * distance = 96 joules

Power required:
Power = Work / 1sec = 96 watts

Please note that the distance moved is the distance that the object would move if it were not canceled by the Earth's gravitational acceleration 9.8 ms^-2.

 

[Dale]

Work done in moving 9.8 meters

You are not moving, you are hovering.

 

[TurtleMeister]

Work done in moving 9.8 meters

You are not moving, you are hovering.

Please note that the distance moved is the distance that the object would move if it were not canceled by the Earth's gravitational acceleration 9.8 ms^-2.

Okay. Just disregard that the object is near the Earth's surface. How much power would be required to accelerate 1kg at 9.8ms^-2 in the absence of any gravitational field? After you've calculated that then place the object near the Earth's surface with it's direction of acceleration pointing away from the earth. What will happen?

 

[Dale]

How much power would be required to accelerate 1kg at 9.8ms^-2 in the absence of any gravitational field?

To accelerate 1 kg at 9.8 m/s² requires 9.8 N of force, not any specific power. In fact, when v=0 m/s the power required is 0 J.

 

[TurtleMeister]

To accelerate 1 kg at 9.8 m/s² requires 9.8 N of force, not any specific power. In fact, when v=0 m/s the power required is 0 J.

So it takes ZERO power for a rocket to hover near the Earth's surface? I guess I'm as mystified as the OP.

Edit: I guess in the case of the rocket the power is used to accelerate the exhaust? So what would that power be for a 1kg rocket? (disregarding decrease in weight of the burning fuel)

 

[Dale]

So it takes ZERO power for a rocket to hover near the Earth's surface?

In the limit of an infinite exhaust mass and an infinitesimal exhaust velocity, yes. (e.g. sit the rocket on a shelf)

 

[stewartcs]

So it takes ZERO power for a rocket to hover near the Earth's surface? I guess I'm as mystified as the OP.

Edit: I guess in the case of the rocket the power is used to accelerate the exhaust?

Objects such as helicopters that are hovering obviously require power even though they are hovering (i.e. the blades must move to generate thrust).

However, you and the OP are missing the point. There is no minimum amount of power required (unless of course you change the object such that it takes advantage of buoyancy or some similar concept). For "hovering" objects such as those described, the amount of power required is dependent on the system.

For example, if we model the system as a fluid jet impinging on a flat plate then the force of the jet (ideally) is equal to the pQV (density x flow rate x velocity). So to get an object to hover based on this model, the force generated by the jet must equal the weight of the object. However, the force generated by the jet is dependent on the density and flow rate (velocity and flow rate are related in this case). In order to get the required flow rate for a given density of fluid there has to be some other device creating the flow rate (typically a pump). The pump requires a power source (typically a motor) which plainly requires power. The amount of power depends on the motor, pump, density of the fluid, etc...

As you can see, there are infinitely many different combination's that will create the required force from the jet and they all may require a different amount of power depending on the design...and that's just one model. Now change the model to a gas turbine engine and investigate all of the parameters required for a certain thrust...now change the model again to something else...hopefully you get the point.

Essentially it becomes a matter of which system is the most efficient.

CS

 

[russ_watters]

Essentially, your error is in the assumption that the amount of air you need to shove downwards remains the same. This is not correct. You can shove more or less air downwards depending on how much you accelerate the air. A small-mass high-velocity air movement will be much more energetic than a large-mass low-velocity air movement.

To expand a little more, force is a change in momentum over time, which is a function of mv. Power is a change in energy over time, which is a function of mv^2. So if you hold force constant by increasing m (mass flow rate) and decreasing v (velocity imparted on the air by the rotor), power goes down. For example, doubling the mass flow rate and halving the velocity cuts the power in half.

 

[TurtleMeister]

Thanks for the info everyone. I see now that thrust cannot be converted to power. However, after doing a little research on the web for model helicopters I've found that the rule of thumb is 50 to 60 watts per pound - just for hovering. It's interesting that this is close to the value that I arrived at. But I'm sure it's just coincidence.

 

[guest1]

Hmm..Ok, I think I get it now (and it bears out mine and a colleague's calculations prior to posting this thread)

Flow rate and velocity imparted on the air should be linked for rotors of constant size - so possibly there is a fixed relation to power if we stipulate a certain area and that the nature of the mass being accelerated should be air
(e.g. by limiting the rotor size or the cross section of the rocket exhaust)

TurtleMeister: I think you miscalculated.
An acceleration of 9.8m/s^2 will not get you to a distance of 9.8 meters in one second

(s = a/2 * t^2)

Your calculation runs up against the same problem as I did in the OP (because it's basically the same approach). If you use the distance traveled in 2 seconds as opposed to one second then you get different power values.