The force acting against a helicopter rotor is drag, not torque. but were going way off topic here.
Jim has his helicopter physics correct. Source: first hand knowledge. Torque is always present so long as the rotor is being powered. I would expect the impeller and turbines in a rocket engine would also impart torque so long as they were being driven by combustion gasses.
That's because you never run a helicopter in a vacuum. If you did, you would still be exerting torque on the rotor, to overcome bearing friction. But the bearings would be exerting the exact same torque back, and the helicoptor would not experience any angular acceleration.
Think of reaction wheels on a spacecraft. When you spin them at constant speed, the spacecraft does not endlessly accelerate, even though the motor has to constantly apply torque to overcome bearing friction. That's because the motor torque (one direction) is exactly equal to the bearing torque (the other direction). They are both referenced to the spacecraft frame. In the helicoptor, one torque is referenced to the air, the other to the helicopter body.
That's because you never run a helicopter in a vacuum. If you did, you would still be exerting torque on the rotor, to overcome bearing friction. But the bearings would be exerting the exact same torque back, and the helicoptor would not experience any angular acceleration.
What? Come on, this is basic Newton's Third Law. If you spin a rotor one way, the helicopter wants to spin the other way. This happens even if the helicopter is in a vacuum. You're committing the same error as the TIME Magazine editor who accused Goddard of not knowing physics.
Think of reaction wheels on a spacecraft. When you spin them at constant speed, the spacecraft does not endlessly accelerate, even though the motor has to constantly apply torque to overcome bearing friction.
Wrong, there are other torques on the spacecraft that cancel this out. Look up magnetorquer
(Sorry, ended up channeling Jim there for a moment...)
Think of reaction wheels on a spacecraft. When you spin them at constant speed, the spacecraft does not endlessly accelerate, even though the motor has to constantly apply torque to overcome bearing friction. That's because the motor torque (one direction) is exactly equal to the bearing torque (the other direction).
Wrong. Spacecraft have multiple reactions wheels in different axis that help cancel out bearing friction torque. A spacecraft with only one RW would spin in the opposite direction. That is why there are 3 or 4 RW in spacecraft.
Honestly, I thought this was high school physics.
A Helicopter has to constantly fight against rotation (with a tail rotor, for example) since the rotor is acting against the atmosphere, and so there is an external torque.
A spacecraft with a rotary component that's internal to itself is almost, but not entirely, absolutely unlike a helicopter. Torque is only applied when the spinning component changes rotational speed. Rev up your car, and if the engine is mounted lengthwise, you'll notice the car tilts. Momentarily. When the RPM changes.
The only way a rocket in steady state can gain spin is if there is a rotational component to the plume that is exiting the closed system. Or, if there's an interaction with the atmosphere. (Which might be affected by the presence of the plume)
Regardless of the physics of helicopters in vacuum, if the torque from the single M1D turbopump was enough to spin up the first stage in the nice thick viscous atmosphere, then the nearly identical turbopump on MVacD should also cause a large roll moment on the second stage during its burn. Why doesn't the second stage also run out of ACS authority, since it should have much less mass/volume margin to carry propellant? Just look at F9 flight 1, where the roll control system (turbopump exhaust gimabling) failed - it took several minutes for even a noticeable roll to build up, and even then the roll never reached a level where it centrifuged the propellant and shut the stage down. So while the turbopump may exert a torque on the rocket, to me it doesn't seem like it would be large enough to cause the spin up the flight 6 stage saw.
Liquid level.
End of the second stage burn should have as little fuel as the first stage had, still no large roll...
The only way a rocket in steady state can gain spin is if there is a rotational component to the plume that is exiting the closed system.
That seems likely without 2 counter-rotating turbopumps...
Just look at F9 flight 1, where the roll control system (turbopump exhaust gimabling) failed - it took several minutes for even a noticeable roll to build up, and even then the roll never reached a level where it centrifuged the propellant and shut the stage down.
Do we know where the GG exhaust got stuck for that? The default position could be offset from center.
I guess the other thing is what sort of spin the stage can pick up from falling through the atmosphere. If it's falling slightly side on and there's imperfections that present more drag from one direction than the other, or something about the engines at the bottom, it could start spinning.
Then there might be propellant in the lines to the engine without any being available near the inlets. That could explain a start without continuing to burn.
A spacecraft with a rotary component that's internal to itself is almost, but not entirely, absolutely unlike a helicopter. Torque is only applied when the spinning component changes rotational speed. Rev up your car, and if the engine is mounted lengthwise, you'll notice the car tilts. Momentarily. When the RPM changes.
The only way a rocket in steady state can gain spin is if there is a rotational component to the plume that is exiting the closed system. Or, if there's an interaction with the atmosphere. (Which might be affected by the presence of the plume)
Thank you for upholding
the law of conservation of angular momentum! Seemed like it was under an attack for a while
OK this is way too long, but I hope I can help clarify some of the misunderstandings around this topic. I think several people on this page are talking past each other as they are making different assumptions. Executive summary: MeekGee is correct, and the pump will not cause the stage to spin because of friction in the pump, but the turbine exhaust could if it is spinning.
For all the examples on this page, you need to construct a control volume around the system being considered. Let's start with the easiest one, the satellite with a single reaction wheel. Let's consider an initial state with both satellite and reaction wheel at rest in an inertial frame, and draw a control volume around the satellite. Let's say the satellite has 100 times the moment of inertia of the reaction wheel. If you decide to spin up the reaction wheel (and you are not using any devices to apply torque to the satellite via momentum exchange across the control volume boundaries with the surroundings, like thrusters), you must apply a torque on the wheel. Such a torque requires an equal and opposite torque to be reacted on the satellite frame. Let's say you apply sufficient torque for sufficient time to develop an angular rate of 100 RPM in the reaction wheel. Because of the relationship of the moments of inertia specified above, the satellite will be rotating at 1 RPM in the opposite direction (relative to the initial inertial frame). The total angular momentum is still conserved (and is zero) because the angular momentum of the reaction wheel cancels that of the satellite.
Now, from this state, let's say that you get some friction in the bearings of the wheel. This will cause the wheel to start to slow down. But the torque applied by the friction has to be reacted by an equal and opposite torque on the satellite's mass, which will also tend to slow the satellite down (since it is spinning in the opposite direction). Given long enough, the torque will bring both bodies to the same angular velocity, namely zero. They will both be stationary just like in the initial condition. The only way you can change this is if you transfer angular momentum out of the control volume, with a device that ejects mass such as a thruster, or one that acts on another body, such as earth, via interaction with the gravitational field gradient or with the magnetic field. This is called desaturating a reaction wheel, when used to reduce the wheel's velocity that has built up (typically after countering an external torque like drag). Applying just enough continuous torque on the reaction wheel to overcome friction will maintain the angular velocity of both stage and wheel indefinitely.
A helicopter at steady state has the same thing going on when considering friction in the bearings, and the motor must apply a certain continuous torque to maintain rotor speed, but this is a very minor effect when considering the external torques on the helicopter. In air, the rotor also experiences drag, which tries to slow the rotor down by transferring angular momentum from the rotor to the air (crossing the control volume), which applies an external torque on the rotor. The motor fights this by applying an opposite torque to the rotor shaft. This torque is reacted in an equal and opposite way on the helicopter body via the motor mounts. In order to prevent the body from spinning the opposite direction from the rotor, the helicopter must generate a torque on the body by transferring angular momentum to the air through lift force from the tail rotor. For a stationary helicopter, the amount of torque generated by the tail rotor is equal to that generated by the rotor, but in opposite directions.
OK, moving on to the first stage. For this part, let's ignore the pump side of the turbopump for a bit and just consider the turbine. Again, if your initial condition is a non-rotating stage and pump, starting up the engine and accelerating the turbine to operating speed (if you could somehow do it without ejecting any mass from your control volume) would require that the angular momentum of the turbine be cancelled by an opposite rotation in the stage, although it would be a small one. Consider a rough estimate: turbine moment of inertia is equivalent to 10 lbs at 6 inches from shaft center, and rotates at 25000 RPM; stage moment of inertia is equivalent to 30000 lbs at 6 ft from stage center. Solving for angular speed (and if I did my math correctly), you will see that the stage would only be rotating at approximately .001 RPM to have equal and opposite angular momentum.
Once the turbine is spun up, though, it cannot cause the stage to spin from viscous drag or bearing friction without also losing speed and transferring some of its angular momentum back to the stage, or providing an external torque. Assuming the the first is not the case as the engine is still running at constant power, this would require an external torque to keep the turbine spinning. The only way to provide an external torque is to eject spinning gas from the turbine. It is possible to do this, but it is also possible (and more likely) that there is a stator element after the turbine to recover the angular momentum and leave the gas without spin as it exits the turbine. If this is done exactly (i.e. absolutely no spin in the ejected gas) then the torque on the stator, that acts back on the stage to try to slow it down, will exactly cancel the torque from friction and viscous drag on the turbine that are trying to spin up the stage in the first place. If it's not done exactly, the stage will spin to conserve angular momentum with the angular momentum that is leaving the control volume with the ejected gas, but you can't tell which way it will spin as you don't know if the gas was under spun or over spun without having intimate knowledge of the SpaceX design.
I'll leave it as an exercise to the reader what happens at the pump end of the turbopump.
A spacecraft with a rotary component that's internal to itself is almost, but not entirely, absolutely unlike a helicopter. Torque is only applied when the spinning component changes rotational speed. Rev up your car, and if the engine is mounted lengthwise, you'll notice the car tilts. Momentarily. When the RPM changes.
The only way a rocket in steady state can gain spin is if there is a rotational component to the plume that is exiting the closed system. Or, if there's an interaction with the atmosphere. (Which might be affected by the presence of the plume)
Thank you for upholding the law of conservation of angular momentum! Seemed like it was under an attack for a while 
And the funny thing is to see who attacked it!
blazotron, thank you for that post!
I was surprised by what some people here were saying and was beginning to question my own understanding of the law of conservation of angular momentum
The only way a rocket in steady state can gain spin is if there is a rotational component to the plume
But while the turbopump is spinning up the rocket is not in a steady state. You said so yourself: "Torque is [...] applied when the spinning component changes rotational speed." That torque
might have been sufficient to slosh the propellant away from the sump.
Spacecraft have multiple reactions wheels in different axis that help cancel out bearing friction torque. A spacecraft with only one RW would spin in the opposite direction. That is why there are 3 or 4 RW in spacecraft.
The reason for 3 RWs is to have full 3 DOF attitude control. 4th is a spare, they can be arranged so that anyone of the four can fail and the remaining three still provide full attitude control.
Nanosat pyramid arrangement:

Tetrahedron arrangement:
Some light reading comparing the systems
The only way a rocket in steady state can gain spin is if there is a rotational component to the plume
But while the turbopump is spinning up the rocket is not in a steady state. You said so yourself: "Torque is [...] applied when the spinning component changes rotational speed." That torque might have been sufficient to slosh the propellant away from the sump.
See blazotron's post on a BOTE calculated RPM that startup would impart to the stage. Also, you're again back to the obvious question on why ACS on 2nd stage is perfectly able to keep the startup torque *undetectable*, but ACS on 1st stage can't cope with it and ends up in a stage rolling so fast it centrifuges propellant. All that within a second or so of engine startup? Not even vastly different propellant loads can account for that huge difference, IMO of course.
The only way a rocket in steady state can gain spin is if there is a rotational component to the plume
But while the turbopump is spinning up the rocket is not in a steady state. You said so yourself: "Torque is [...] applied when the spinning component changes rotational speed." That torque might have been sufficient to slosh the propellant away from the sump.
See blazotron's post on a BOTE calculated RPM that startup would impart to the stage. Also, you're again back to the obvious question on why ACS on 2nd stage is perfectly able to keep the startup torque *undetectable*, but ACS on 1st stage can't cope with it and ends up in a stage rolling so fast it centrifuges propellant. All that within a second or so of engine startup? Not even vastly different propellant loads can account for that huge difference, IMO of course.
Agree with you, and to add something more, transferring the roll to the propellant in such a short time (bold is mine) requires something more than some vortex baffles on the sump.
People, this is over: aero torque...
Honestly, I thought this was high school physics.
A Helicopter has to constantly fight against rotation (with a tail rotor, for example) since the rotor is acting against the atmosphere, and so there is an external torque.
A spacecraft with a rotary component that's internal to itself is almost, but not entirely, absolutely unlike a helicopter. Torque is only applied when the spinning component changes rotational speed. Rev up your car, and if the engine is mounted lengthwise, you'll notice the car tilts. Momentarily. When the RPM changes.
The only way a rocket in steady state can gain spin is if there is a rotational component to the plume that is exiting the closed system. Or, if there's an interaction with the atmosphere. (Which might be affected by the presence of the plume)
There is friction in the internal components therefore there is constant torque and transfer of momentum. Example, the cage around a toy gyroscope ends up spinning with the gyro rotor.
0:26 into the video
Honestly, I thought this was high school physics.
A Helicopter has to constantly fight against rotation (with a tail rotor, for example) since the rotor is acting against the atmosphere, and so there is an external torque.
A spacecraft with a rotary component that's internal to itself is almost, but not entirely, absolutely unlike a helicopter. Torque is only applied when the spinning component changes rotational speed. Rev up your car, and if the engine is mounted lengthwise, you'll notice the car tilts. Momentarily. When the RPM changes.
The only way a rocket in steady state can gain spin is if there is a rotational component to the plume that is exiting the closed system. Or, if there's an interaction with the atmosphere. (Which might be affected by the presence of the plume)
If there is friction in the internal component then there is constant torque and transfer of momentum. The cage around a toy gyroscope ends up spinning with the gyro rotor.
I agree that's what the gyroscope does, but it's because the rotor is slowing down. If you add a motor, attached to the frame, that keeps the rotor at constant speed, then the cage will not rotate (after startup). There is indeed torque, in fact there are two torques, and they cancel. The rotor drag tries rotate the frame, but the motor applies an exactly opposite torque on the frame, while doing its work to keep the rotor at constant speed. Net result - no rotation of the cage.