Problem 25 - University Physics with Modern Physics

Question

A figure skater draws her arms in during a final spin. Since angular momentum is conserved, her angular velocity will increase. Is her rotational kinetic energy conserved during this process? If not, where does the extra energy come from or go to?

Step-by-Step Solution

Answer

Answer: No, a figure skater's rotational kinetic energy is not conserved when she pulls her arms in during a final spin. The extra energy comes from the work done by the skater's muscles in pulling her arms towards her body, which increases her angular velocity due to the conservation of angular momentum.

Step 1: Understand the Conservation Law of Angular Momentum

According to the conservation law of angular momentum, the total angular momentum of a closed system remains constant. In this case, when the figure skater pulls her arms in, her moment of inertia decreases. Consequently, her angular velocity increases to keep the angular momentum constant. Mathematically, this can be written as: I_1ω_1 = I_2ω_2

Step 2: Define Rotational Kinetic Energy

Rotational kinetic energy is the kinetic energy related to an object's rotation. It can be calculated using the following formula: K = 0.5 * I * ω²

Step 3: Is Rotational Kinetic Energy Conserved?

Now, we will analyze whether rotational kinetic energy is conserved during this process or not. Let's denote the initial and final rotational kinetic energies as K_1 and K_2, respectively: K_1 = 0.5 * I_1 * ω_1^2 K_2 = 0.5 * I_2 * ω_2^2 Since ω_2 > ω_1, it is clear that K_2 > K_1, which means the rotational kinetic energy is not conserved and has increased during the process.

Step 4: Identifying the Source of Extra Energy

To maintain the conservation of energy, the increase in rotational kinetic energy must come from another form of energy. In this case, the figure skater does work to pull her arms in towards her body, converting her internal muscular energy into rotational kinetic energy. This increase in energy can also be viewed as a decrease in gravitational potential energy as the skater is no longer supporting her arms against gravity. In conclusion, the figure skater's rotational kinetic energy is not conserved during the process of pulling her arms in. Instead, the extra energy comes from the work done by the skater's muscles in pulling her arms towards her body, which increases her angular velocity as a result of the conservation of angular momentum.