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Why is the gyroscopic effect going to be a problem with a flywheel on a stationary platform? If anything it will make the trailer more stable, wouldn't it?
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Well the trailer is not going to be stationary once the car/trailer is moving. If you suddenly drive onto a section of the road that has a sideways tilt the trailer will want to stay level, i.e. pick a tire up off the ground. Also as you add and remove energy to the flywheel (speed it up or slow it down) you will have the trailer want to move in an equal and opposite direction.