# Fouette Turns

I used to be a ballet dancer, and I remember that one of the hardest skills to learn and master was the fouette turn. This move requires the ballerina to extend her leg out and then bring it quickly in, all while completing a turn (see diagram below for clarity). These turns can go on forever, and the secret behind their infinite completion is all based in the idea of conservation of angular momentum. Angular momentum is equal to the moment of inertia (I) multiplied by omega (w). Moment of inertia is determined by the integral of the radius squared multiplied by dm; because the mass of the dancer is unchanging during the turn sequence, the only factor that will influence her moment of inertia is her radius. So, when the dancer begins her turn sequence, she has her leg extended out in front of her, causing the radius of her body to be larger and therefore her moment of Inertia to increase; consequently, at this point in the turn, her angular velocity is rather low. However, when she whips her leg in, her radius decreases, causing her moment of Inertia to decrease proportionally. Because of the law of conservation of angular momentum, the angular momentum at the start of the turn must equal the angular momentum when the dancer whips her leg in. Because angular momentum = Iw, a decrease in I will result in an increase in w and vice versa in order to fulfill the law of conservation of angular momentum. So, when the dancer whips her leg in, causing her moment of Inertia to decrease, her angular velocity must increase in order to compensate and conserve momentum; the increased angular velocity allows the dancer to complete a revolution and extend her leg out again and so and so forth. Hence, these turns can go on forever because angular momentum must be conserved; the only way to stop them is to provide a net torque to stop the rotation, or, more often then not, to lose your balance and fall over!

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