Competitive ice skaters commonly perform single, double, and triple axel jumps in which they rotate $1\frac{1}{2}$, $2\frac{1}{2}$ and $3\frac{1}{2}$ revolutions, respectively, about a vertical axis while airborne. For each of these jumps, a typical skater remains airborne for about 0.70 s. Suppose a skater leaves the ground in an “open” position (e.g., arms outstretched) with moment of inertia I₀ and rotational frequency ƒ₀ = 1.2 rev/s , maintaining this position for 0.10 s. The skater then assumes a “closed” position (arms brought closer) with moment of inertia I, acquiring a rotational frequency f, which is maintained for 0.50 s. Finally, the skater immediately returns to the “open” position for 0.10 s until landing (see Fig. 11–51). Why is angular momentum conserved during the skater’s jump? Neglect air resistance.
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1. Angular momentum is conserved because there are no external torques acting on the skater while airborne. This is a direct consequence of the law of conservation of angular momentum, which states that in the absence of external torques, the total angular momentum of a system remains constant.
2. The skater's angular momentum can be expressed as L = Iω, where I is the moment of inertia and ω is the angular velocity. Since angular momentum is conserved, the product of I and ω must remain constant throughout the jump.
3. Initially, the skater is in an 'open' position with a moment of inertia I₀ and an angular velocity ω₀. The angular velocity ω₀ can be calculated from the given rotational frequency ƒ₀ using the relationship ω₀ = 2πƒ₀.
4. When the skater transitions to the 'closed' position, the moment of inertia decreases to I. To conserve angular momentum, the angular velocity must increase, such that I₀ω₀ = Iω. This relationship allows us to calculate the new angular velocity ω in the 'closed' position.
5. Finally, the skater returns to the 'open' position before landing. The moment of inertia increases back to I₀, and the angular velocity decreases accordingly to maintain conservation of angular momentum. This ensures that the skater's total angular momentum remains constant throughout the jump.

Key Concepts

Angular Momentum

Angular momentum is a physical quantity that represents the rotational inertia and rotational velocity of an object. It is defined as the product of an object's moment of inertia and its angular velocity. In a closed system, angular momentum remains constant unless acted upon by an external torque, which is crucial for understanding the skater's jumps.

Moment of Inertia

Moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on the mass distribution relative to the axis of rotation; the further the mass is from the axis, the greater the moment of inertia. In the context of the skater, changing positions (open to closed) alters the moment of inertia, affecting the rotational speed.

Conservation of Angular Momentum

The conservation of angular momentum states that if no external torques act on a system, the total angular momentum of that system remains constant. For the skater, as she transitions between open and closed positions, her moment of inertia changes, but to conserve angular momentum, her rotational frequency must adjust accordingly, allowing her to complete the required revolutions during the jump.