# Can lift contradict conservation of energy?

In cruise flight of an aircraft lift does no work, it just holds the aircraft aloft. In order to climb aircraft pitch up and need excess energy because the thrust vector now has a vertical component acting against gravitational acceleration.

Please take a look at the following cart. The thrust gets delivered by a propeller at the rear and the thrust vector is always horizontal. A wing is attached to a vertical pole.

An experiment is done with the following steps:

The cart is at rest and the wing is locked to the pole. Then the cart is accelerated and when it reaches terminal velocity the engine is turned off and the locking of the wing is released at the same time.

When the lift produced by the wing is greater than its weight, the wing begins to accelerate vertically. At some point the wing reaches a vertical terminal velocity at which further acceleration stops. The drag onto the cart is greater when the wing moves upwards compared to when it does not.

The drag slows down the cart. The kinetic energy of the cart is converted into thermal energy (some of the cart’s kinetic energy is transferred onto the surrounding air, but in the end all dissipates to thermal energy).

Conclusion: The kinetic energy of the cart is converted entirely into thermal energy. No kinetic energy of the cart goes into potential energy of the wing, the movement of the wing is done purely by lift.

Did I miss something?

• That's some fancy graphic design. – Harry Wilson Jun 12 '15 at 12:23
• Why don't you just refer to a balloon? – John Duffield Jun 12 '15 at 13:07
• There's a disconnect between wondering whether the air is doing work on the vertically-mounted wing, but assuming that the rotary-mounted wings on the propeller have driven the cart. – rob Nov 28 at 16:48

When the lift produced by the wing is greater than its weight, the wing begins to accelerate vertically.

I can see where you're coming from - essentially questioning the physicality of wing lift on the basis of thermodynamics. In a simple conception of the situation, this does seem like a problem.

But let's look more closely at the gravitational potential energy of the wind and the power imparted by the wind. If the wing is moving upward at $v$, then the power imparted by the wind is $Fv$, and ultimately $mgv$ in a steady-state scenario (where m is the mass of the wing).

The upward velocity of the wing comes with a "gotcha". As the wing moves upward, the wind is still lateral. That makes its angle-of-attack rise above the horizontal. We don't need to assume any lift-to-drag ratio. Even if we insert a perfect wing, the act of climbing (and increasing gravitational potential) will include a component of the force pushing the cart backwards. The cart is moving forward with stored energy applied by the experimenters, and this force saps that energy.

I would be interested to see if there is a physical upper bound on a lift-to-drag ratio. However, I don't think that such a thing exists. This particular thought experiment (at least) doesn't reveal one.

• There may be no physical limit on lift-to-drag ratio for perfect fluids, but for viscous fluids making the wings larger with ever smaller angles of attack is a losing proposition, at least for wings that are made from real matter and not angle-feathers (which can be infinitely thin, yet are infinitely strong). – CuriousOne Jun 12 '15 at 19:06
• @Alan Rominger: The relative wind (as seen from the perspective of the wing) changes when the wing moves up. Since lift is perpendicular to relative wind, there is an additional lift-induced drag onto the cart when the wing accelerates up. Are you referring to this? – Chris Jun 13 '15 at 8:10
• @Chris Yes, that is what I was claiming in very clear terms. Obviously I haven't ran the math, but I suspect that this effect alone can rule out the perpetual motion possibility. – Alan Rominger Jun 13 '15 at 11:45

Any aircraft has potential energy due to its altitude, and kinetic energy due to its velocity. The sum of these is its total energy. If the stick is pushed forward or back, the aircraft simply trades potential energy for kinetic energy or vice-versa, exactly like an earthbound roller-coaster.

In order to descend to a landing, the aircraft must dissipate some of this energy, and the only way to do that is with drag, of which there are two kinds, parasitic and induced. Parasitic drag is just the "rubbing" of the air due to its viscosity. Induced drag is due to the fact that the lift vector is not vertical, it is slanted backward, and the horizontal component is to the rear.

Efficient aircraft can have very low drag, but this has a disadvantage. It's hard to come down. If the drag were zero, they could stay up forever.

The next time you travel in a commercial airliner, take note of the rectangular spoilers on the tops of the wings. The purpose of these is to 1) create drag when raised a little, and 2) cut lift when raised a lot. They are also called "speed brakes". The pilot may use these to increase descent rate. (They are also used when the wheels touch down, to make the wings stop lifting.)

You mention nothing about the drag on the airplane wing, and although I'm no aerodynamicist I'm pretty sure that an airfoil can't produce lift without producing drag. Add in drag, and your apparent perpetual flight machine fails.

Ignore the propeller: think about a glider. Gliders always have a glide slope: they can never maintain speed at the same time as they maintain altitude (ignoring thermals and the like). Yes, you can pull back on the stick and they magically go up, but you lose speed both from drag and from the fact that the airfoil's lift is no longer straight up, and if you don't stop after a while you'll fall from the sky.

Or, go all-in on the propeller and think about a hovering helicopter. It takes a lot of energy for them to just stay put, energy that's dissipated into the column of air being blown down.