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Chapter 10: Problem 14

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

Expert verified
a) Her kinetic energy of rotation remains constant. 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 her getting tired and the decrease in rotational inertia.

Step by step solution

01

Understand the Concepts

To answer this question correctly, it's essential to understand three primary concepts: angular momentum (L), rotational inertia (I), and the kinetic energy of rotation (K). Angular Momentum (L) is a measure of the rotation of an object, given by the product of its rotational inertia (I) and its angular velocity (蠅). Mathematically, it can be represented by: L = I蠅 Rotational Inertia (I) is a measure of an object's resistance to change in its rotation, which depends on the distribution of mass and the axis of rotation. The Kinetic Energy of Rotation (K) is the energy associated with the rotatory movement of an object, given by the formula: K = (1/2)I蠅^2 The conservation of angular momentum states that the total angular momentum of a closed system does not change.
02

Apply the Conservation of Angular Momentum

When the ice skater pulls her arms in, her rotational inertia (I) decreases because her mass is closer to the axis of rotation. According to the conservation of angular momentum, since L is conserved, the decrease in I must cause an increase in her angular velocity (蠅). Let I1 and 蠅1 be her rotational inertia and angular velocity when her arms are extended, and I2 and 蠅2 when she pulls her arms in. Then, applying the conservation ofangular momentum, we have: I1蠅1 = I2蠅2 Since I2 < I1 (rotational inertia decreases), it must be the case that 蠅2 > 蠅1 (angular velocity increases).
03

Analyze the Kinetic Energy of Rotation

Now that we've established that her angular velocity (蠅) increases, let's examine how her kinetic energy of rotation (K) changes. As mentioned earlier, the formula for the kinetic energy of rotation is: K = (1/2)I蠅^2 Initially, her kinetic energy of rotation is: K1 = (1/2)I1蠅1^2 After pulling her arms in, her kinetic energy of rotation becomes: K2 = (1/2)I2蠅2^2 Since both I2 < I1 and 蠅2 > 蠅1, it is not immediately clear whether K2 > K1 or K2 < K1.
04

Determining the Correct Statement

Now let's analyze each statement. a) False. While her angular momentum is conserved, it does not mean that her kinetic energy of rotation remains constant. b) True. When the skater pulls her arms in, she does work to overcome the rotational inertia. This work is converted into kinetic energy, causing her kinetic energy of rotation to increase. c) False. Her kinetic energy of rotation does not decrease due to her getting tired or the decrease in her rotational inertia. Instead, it increases because of the work she does to pull her arms in. So, the correct statement is (b) Her kinetic energy of rotation increases because of the work she does to pull her arms in.

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