Skater's Arms In: Angular Momentum Conservation

TITLE: Skater's Arms In: Angular Momentum Conservation DESCRIPTION: Understand why a spinning skater's angular velocity increases when they pull their arms inward, explained through the principle of conservation of angular momentum.

Concept Overview

This question explores a fundamental principle in rotational motion: the conservation of angular momentum. When there is no external net torque acting on a system, its total angular momentum remains constant. By pulling their arms inward, a skater changes their moment of inertia, and to conserve angular momentum, their angular velocity must increase. This demonstrates the inverse relationship between moment of inertia and angular velocity for a constant angular momentum.

Step 1: Define Angular Momentum Angular momentum ($L$) for a point mass is given by the product of its moment of inertia ($I$) and its angular velocity ($omega$). For a rigid body or a system of particles, the total angular momentum is the sum of the angular momenta of its constituent parts, or more generally, $L = Iomega$.

This equation states that angular momentum is a measure of an object's rotational inertia and its speed of rotation.

Step 2: State the Condition for Conservation of Angular Momentum Angular momentum is conserved if and only if the net external torque ($tau_{ext}$) acting on the system is zero. The relationship between torque and angular momentum is given by:

If $tau_{ext} = 0$, then $dL/dt = 0$, which implies that $L$ is constant.

Step 3: Analyze the Skater System Consider a spinning ice skater as a system. When the skater is spinning with arms extended, their mass is distributed farther from the axis of rotation. When the skater pulls their arms inward, their mass is brought closer to the axis of rotation.

Step 4: Define Moment of Inertia The moment of inertia ($I$) is a measure of an object's resistance to changes in its rotational motion. It depends on the mass of the object and how that mass is distributed relative to the axis of rotation. For a system of particles, it is generally given by $I = \sum m_i r_i^2$, where $m_i$ is the mass of the $i$-th particle and $r_i$ is its perpendicular distance from the axis of rotation.

When the skater pulls their arms inward, the average distance ($r_i$) of the mass from the axis of rotation decreases, thus decreasing the skater's total moment of inertia ($I$).

Step 5: Apply Conservation of Angular Momentum As the skater spins, we can assume that the net external torque acting on them is negligible (friction from the ice and air resistance are small). Therefore, their angular momentum ($L$) is conserved.

Using the definition of angular momentum ($L = Iomega$), we can write:

Step 6: Explain the Increase in Angular Velocity Initially, the skater's arms are extended, resulting in a larger moment of inertia ($I_{initial}$). When the skater pulls their arms inward, the moment of inertia decreases ($I_{final} < I_{initial}$). Since angular momentum ($L$) must remain constant, and $L = Iomega$, if $I$ decreases, then $omega$ must increase to compensate.

Since $I_{initial} > I_{final}$, the ratio $I_{initial}/I_{final}$ is greater than 1, meaning $omega_{final} > omega_{initial}$. Thus, the skater spins faster.

Key Takeaways:

• Angular momentum ($L = Iomega$) is conserved in the absence of external torque.
• Moment of inertia ($I$) depends on mass distribution; bringing mass closer to the axis of rotation decreases $I$.
• When $I$ decreases and $L$ is conserved, angular velocity ($omega$) must increase.
• This principle is a direct application of the conservation of angular momentum.

Answer: The angular velocity increases because the skater's moment of inertia decreases when they pull their arms inward, and angular momentum ($L = I\omega$) must be conserved, leading to a higher angular velocity ($omega$).