Problem 15 - University Physics with Modern Physics

Question

An ice skater rotating on frictionless ice brings her hands into her body so that she rotates faster. Which, if any, of the conservation laws hold? a) conservation of mechanical energy and conservation of angular momentum b) conservation of mechanical energy only c) conservation of angular momentum only d) neither conservation of mechanical energy nor conservation of angular momentum

Step-by-Step Solution

Answer

Answer: Conservation of mechanical energy and conservation of angular momentum.

Step 1: Understand the conservation of mechanical energy

Conservation of mechanical energy states that if only conservative forces are acting on a system, the total mechanical energy of the system remains constant. In this case, the ice skater is on frictionless ice, so there are no non-conservative forces (like friction or air resistance) acting on her. Thus, the mechanical energy remains constant.

Step 2: Understand the conservation of angular momentum

The conservation of angular momentum states that the total angular momentum of a system remains constant if no external torques act upon it. In this case, the ice skater is on a frictionless surface, and no other external forces are applied, so there are no external torques. Thus, the angular momentum remains constant as well.

Step 3: Apply the conservation laws to the ice skater's motion

The ice skater's mechanical energy and angular momentum are conserved as she brings her hands into her body, as there are no external non-conservative forces or external torques acting upon her. When she brings her hands closer to her body, her moment of inertia decreases, while her angular velocity increases, resulting in a faster spin. However, the mechanical energy remains constant as the kinetic energy of her spinning motion is not changed due to internal redistribution of mass. Similarly, the total angular momentum remains constant as the product of her moment of inertia and angular velocity remains the same.

Step 4: Choose the correct answer

Based on the analysis in Steps 1-3, it is clear that both conservation of mechanical energy and conservation of angular momentum hold for this problem. Therefore, the correct answer is: a) conservation of mechanical energy and conservation of angular momentum