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Chapter 10: Problem 15

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

Short Answer

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

Step by step solution

01

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.
02

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.
03

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.
04

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

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Most popular questions from this chapter

A uniform solid cylinder of mass \(M=5.00 \mathrm{~kg}\) is rolling without slipping along a horizontal surface. The velocity of its center of mass is \(30.0 \mathrm{~m} / \mathrm{s}\). Calculate its energy.

A ball attached to the end of a string is swung in a vertical circle. The angular momentum of the ball at the top of the circular path is a) greater than the angular momentum at the bottom of the circular path. b) less than the angular momentum at the bottom of the circular path. c) the same as the angular momentum at the bottom of the circular path.

A solid sphere rolls without slipping down an incline, starting from rest. At the same time, a box starts from rest at the same altitude and slides down the same incline, with negligible friction. Which object arrives at the bottom first? a) The solid sphere arrives first. b) The box arrives first. c) Both arrive at the same time. d) It is impossible to determine.

Does a particle traveling in a straight line have an angular momentum? Explain.

A sphere of radius \(R\) and mass \(M\) sits on a horizontal tabletop. A horizontally directed impulse with magnitude \(J\) is delivered to a spot on the ball a vertical distance \(h\) above the tabletop. a) Determine the angular and translational velocity of the sphere just after the impulse is delivered. b) Determine the distance \(h_{0}\) at which the delivered impulse causes the ball to immediately roll without slipping.
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