Conservation of angular momentum while sitting on a spinning chair Today my friend was sitting on a spinning char. By moving his top part of the body left to right and his bottom part of the body the opposite he managed to spin. As I understand Conservation of angular momentum if he was just sitting still he should not be able to spin. He has not touched anything (wall, table). The question is how has he changed his angular momentum (and the angular momentum of the chair) just by moving his body while sitting on the chair.
 A: The angular momentum is conserved only if the net torque on a system is zero. By performing his body motions, your friend was able to let the ground generate a torque on him, by the virtue of the frictional force acting on the chair due to sideways movement. If no frictional force were there, your friend would never spin. What happens is your friend pushed the chair in such a way that the frictional force acting on the chair induced a torque on it, so while the magnitude of the force equalled the force exerted, the point at which it was applied was different, generating a torque. That's why the conservation of angular momentum seemed to be violated. The total angular momentum of the chair-earth system of course remains conserved as earlier.
A: You do not say, but I assume his feet were in the air and not on the floor which is why you are wondering about conservation of angular momentum.
I suspect the spin axis of the chair was not exactly vertical and the plane of all possible positions was tilted from the horizontal, one side of this plane is actually lower and the other is actually higher. So, when he shifted his body rearranging it side wise and changed the position of his body he was actually moving the heaviest portion of his body from the lower side to the higher side. In doing so he raised his center of gravity and did work. As the heavier side of his body was now on the higher side of the plane of spin, the chair rotated a half turn returning the heavier side of his body to the lower position on the plane of possible positions. He was sliding down hill on a tilted  circular track. The chair/human system was returning to its lower energy state. 
Depending on friction etc his position may have over shot and oscillated to some extent, but after time it will settle with the center of gravity in the lowest permitted position. Once the CG is at the lowest permitted position, the system is in the lowest possible energy state and will remain there.
If he continues to shift his position with each half turn of rotation, he can continue to add energy to the system and repeatedly move it from its lowest energy position causing a restoring force/torque to be generated that will continue to rotate the chair/person system.
