# No need Power to fly ? Paradox [duplicate]

I would like to know how much power I need for magnetic levitation.

$$P = Fv.$$

$F = F_g \approx \text{mass} \times 10$, okay. But $v = 0$! I can't believe I need no power to keep an object in the air. How does one explain this?

There is a confusion of what that $v$ means. You are thinking about the velocity of the drone, which is stationary this $v_{drone}=0$. But in your equation, to calculate the power needed by the wings, you have to consider the velocity of the motor providing the thrust (propelling air downwards at a certain rate) to keep the drone floating which is making work.

The speed of the rotors to keep the drone floating and fight back gravity will depend on the mass of it, the size of the wings attached to the rotor, and to the number of them (to keep it simple).

Now, if you want to move around, let's say up, you have to provide an excess of thrust to provide an acceleration at the expense of extra work.

• What about magnetic levitation? – Thomas Sep 28 '15 at 15:37

The problem is that finite wings inherently induce drag (called lift induced drag). Even if you could assume the wing infinite you still have parasitic drag. So you will always need to supply energy to keep the plane flying at a constant speed (and as a consequence to keep generating lift). Without drag it could, in principle, keep flying without power. It's the same with cars for example.

By the way the force that you are using in that equation should be the forward force (thrust) that compensates the drag, not the lift force, since lift is perpendicular to the velocity.

You could read this if you want a simple approximation to drag under certain conditions.

Power equals work/time. Work is force times distance, so you can simplify to force times speed, when the force is constant and the force is causing the speed. In your situation there is a force due to gravity that would do work on your drone, and what you need is for the drone to do work to counter balance that.

In other words, power is always work over time (rate at which work is done), and is force times speed only under the above conditions, which aren't met by your scenario.

This is a good question, because plugging into the formula doesn't work, and it's reasonable to want to know why.

I think this formula would work better F=MxA Since you are hovering you need to consider the acceleration of gravity in your calculations. So you would get F= M x 9.832 meters per second squared. Use kilograms for the mass in this case

In principle, you are right that it is possible to keep an object in the air without expending any energy. Think about a blimp, you can fill it with helium, and then just let it float. It doesn't have to do anything to float, it just floats because it is lighter than air at ground level (the air gets less dense as you go up).

You can even make heavier than air objects float without expending energy. One way to do this is to make a diamagnetic material levitate with magnetic fields. The classic demonstration of this is the levitating frog. You should be able to find videos on youtube.

Now your drone will expend energy, but that is ok. You have shown that it is possible to float without expending energy, not will never expend energy if you float.

• hmmm I get the idea but I feel it answers only partially. What energy will I need for a mass of m and density d to keep it floating in the air ? (d > air) – Thomas Aug 18 '15 at 4:21
• actually if you do nothing you just fall, so you cannot float in the air without expending energy – Thomas Aug 18 '15 at 6:02
• @Thomas No, a blimp really can float with doing anything. Here is a picture of a blimp in case there was any confusion about what a blimp was. Also the blimp was just supposed to be one example of how something can float without expending energy. It's not the only way. I have edited my answer so the second paragraph gives an example of levitating something denser than air without using energy. – Brian Moths Aug 18 '15 at 16:56