Question

An ice skater spins with her arms extended and then pulls her arms in and spins faster. Which statement is correct? a) Her kinetic energy of rotation does not change because, by conservation of angular momentum, the fraction by which her angular velocity increases is the same as the fraction by which her rotational inertia decreases. b) Her kinetic energy of rotation increases because of the work she does to pull her arms in. c) Her kinetic energy of rotation decreases because of the decrease in her rotational inertia; she loses energy because she gradually gets tired.

Short answer

a) Her kinetic energy of rotation does not change due to the conservation of angular momentum. b) Her kinetic energy of rotation increases because of the work she does to pull her arms in. c) Her kinetic energy of rotation decreases due to the decrease in rotational inertia and the skater getting tired. Answer: b) Her kinetic energy of rotation increases because of the work she does to pull her arms in.

Step-by-step solution

Understand the conservation of angular momentum

Conservation of angular momentum states that, in the absence of external torques, the total angular momentum of a system remains constant. As the skater pulls her arms in, her rotational inertia (moment of inertia) decreases, so her angular velocity must increase to keep the total angular momentum constant.

Study the rotational kinetic energy

The rotational kinetic energy of an object is given by the formula \(K = \frac{1}{2}I\omega^2\), where \(K\) is the kinetic energy, \(I\) is the moment of inertia (rotational inertia), and \(\omega\) is the angular velocity. Since the skater's angular velocity increases when she pulls her arms in, we need to determine how this affects her rotational kinetic energy.

Analyze the given statements

Let's analyze the given statements: a) This statement claims that the kinetic energy of rotation does not change due to the conservation of angular momentum. However, this fact directly affects the angular velocity, not the kinetic energy. So, this statement is incorrect. b) This statement claims that the kinetic energy of rotation increases because of the work done to pull her arms in. Since the skater is doing work to pull her arms in, some of this energy must be transferred to her rotational kinetic energy. This statement is correct. c) This statement claims that the kinetic energy of rotation decreases due to the decrease in rotational inertia and the skater getting tired. The skater getting tired is not a reason related to energy conservation, and a decrease in rotational inertia alone does not determine the change in kinetic energy. So, this statement is incorrect.

Conclusion

Based on the analysis, the correct statement is b) Her kinetic energy of rotation increases because of the work she does to pull her arms in.

Key concepts

Rotational Kinetic Energy

Imagine a spinning figure skater as a living example of physics in motion. Rotational kinetic energy is the energy an object possesses due to its rotation. It is represented by the formula \(K = \frac{1}{2}I\omega^2\), where \(K\) stands for kinetic energy, \(I\) is the object's moment of inertia, and \(\omega\) symbolizes angular velocity.

When our skater extends her arms, her moment of inertia is greater, which means she needs more energy to maintain the same rotational speed. Pulling her arms in, her moment of inertia decreases but due to the conservation of angular momentum—think of it as the 'quantity' of rotation that must be preserved—her angular velocity surges, spinning her progressively faster.

In the process of drawing in her arms, the skater does work on her body; this work translates into an increase in her rotational kinetic energy. Thus, the skater, with a boost in angular velocity and a new distribution of mass closer to her axis of rotation, experiences an uplift in rotational kinetic energy.

Angular Velocity

Angular velocity quantifies how fast an object rotates or spins. It is frequently symbolized by \(\omega\) and measured in radians per second (rad/s). Our skater's angular velocity changes dynamically as she pulls her arms closer to her body.

The conservation of angular momentum is key here: it tells us that even without external forces or torques, the skater's angular momentum stays unchanged. What changes is how that momentum is distributed. By tucking in her arms, she decreases her moment of inertia, leading to an increase in angular velocity to compensate.

This elegant dance of physical quantities ensures that when the skater's arms are drawn in, the increase in angular velocity will offset any loss in moment of inertia. The result is a dazzlingly quick spin, seemingly defying gravity, while actually obeying the strict laws of physics.

Moment of Inertia

Now let's dissect the concept of the moment of inertia, which can be visualized as a measure of an object's reluctance to change its rotational motion. Think of it as rotational mass, the rotational equivalent to mass in linear motion.

The moment of inertia depends on both the mass of the object and the distribution of that mass relative to the axis of rotation. With our skater's arms extended, the mass is distributed away from the axis of rotation, granting her a higher moment of inertia. Once she draws her arms in, the mass distribution is closer to the axis, resulting in a lower moment of inertia.

This is crucial in understanding the skater's performance. A lower moment of inertia means it's easier to spin fast. The conservation of angular momentum ensures that as the skater's moment of inertia shrinks, her angular velocity must grow, otherwise, the 'balance' of rotational motion would be broken. It's a perfect demonstration of how these seemingly abstract principles play out in the physical world.