During ice ballet, while in the outer rounds, why do the dancers outstretch their arms and legs.

This is a fairly long answer, but I thought it would be fun to analyze the skater thinking solely about forces. Angular momentum comes in at the end, when it pops up inevitably. I'll give a qualitative answer describing the forces in the system and how they cause spinning, then a quantitative answer to calculate the spin rate. Qualitative Answer As an ice skater pulls in her arms a legs, her arms and legs exert a Set a plate in front of you. Try pushing on at in various places and in various directions. You can imagine an arrow pointing in the direction of the force, starting from the point where you're pushing. If this arrow points towards the center of the plate, the plate will not rotate. Otherwise, it will. In this picture, the red arrows indicate forces on the gray plate. The force labeled "Spin" will induce some spinning in the plate (in addition to accelerating it as a whole) because the dotted line "j" of that force does not go through the center of the plate. The force labeled "NoSpin" will only accelerate the plate and not cause any spinning because it lies on a line that passes through the center of the plate. There is nothing special about the circular plate in this example. Any other shape would work as well, but you would need to define the center point by the To see that the skater spins up as she pulls in her arms, we'll need to find the forces exerted on her body while she pulls her arms in. Then we can see whether these forces point straight towards the c...

This is a fairly long answer, but I thought it would be fun to analyze the skater thinking solely about forces. Angular momentum comes in at the end, when it pops up inevitably. I'll give a qualitative answer describing the forces in the system and how they cause spinning, then a quantitative answer to calculate the spin rate. Qualitative Answer As an ice skater pulls in her arms a legs, her arms and legs exert a Set a plate in front of you. Try pushing on at in various places and in various directions. You can imagine an arrow pointing in the direction of the force, starting from the point where you're pushing. If this arrow points towards the center of the plate, the plate will not rotate. Otherwise, it will. In this picture, the red arrows indicate forces on the gray plate. The force labeled "Spin" will induce some spinning in the plate (in addition to accelerating it as a whole) because the dotted line "j" of that force does not go through the center of the plate. The force labeled "NoSpin" will only accelerate the plate and not cause any spinning because it lies on a line that passes through the center of the plate. There is nothing special about the circular plate in this example. Any other shape would work as well, but you would need to define the center point by the To see that the skater spins up as she pulls in her arms, we'll need to find the forces exerted on her body while she pulls her arms in. Then we can see whether these forces point straight towards the c...