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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

In a tire-throwing competition, a man holding a \(23.5-\mathrm{kg}\) car tire quickly swings the tire through three full turns and releases it, much like a discus thrower. The tire starts from rest and is then accelerated in a circular path. The orbital radius \(r\) for the tire's center of mass is \(1.10 \mathrm{~m},\) and the path is horizontal to the ground. The figure shows a top view of the tire's circular path, and the dot at the center marks the rotation axis. The man applies a constant torque of \(20.0 \mathrm{~N} \mathrm{~m}\) to accelerate the tire at a constant angular acceleration. Assume that all of the tire's mass is at a radius \(R=0.35 \mathrm{~m}\) from its center. a) What is the time, \(t_{\text {throw }}\) required for the tire to complete three full revolutions? b) What is the final linear speed of the tire's center of mass (after three full revolutions)? c) If, instead of assuming that all of the mass of the tire is at a distance \(0.35 \mathrm{~m}\) from its center, you treat the tire as a hollow disk of inner radius \(0.30 \mathrm{~m}\) and outer radius \(0.40 \mathrm{~m}\), how does this change your answers to parts (a) and (b)?

A sheet of plywood \(1.3 \mathrm{~cm}\) thick is used to make a cabinet door \(55 \mathrm{~cm}\) wide by \(79 \mathrm{~cm}\) tall, with hinges mounted on the vertical edge. A small 150 - \(\mathrm{g}\) handle is mounted \(45 \mathrm{~cm}\) from the lower hinge at the same height as that hinge. If the density of the plywood is \(550 \mathrm{~kg} / \mathrm{m}^{3},\) what is the moment of inertia of the door about the hinges? Neglect the contribution of hinge components to the moment of inertia.

A uniform solid sphere of mass \(M\) and radius \(R\) is rolling without sliding along a level plane with a speed \(v=3.00 \mathrm{~m} / \mathrm{s}\) when it encounters a ramp that is at an angle \(\theta=23.0^{\circ}\) above the horizontal. Find the maximum distance that the sphere travels up the ramp in each case: a) The ramp is frictionless, so the sphere continues to rotate with its initial angular speed until it reaches its maximum height. b) The ramp provides enough friction to prevent the sphere from sliding, so both the linear and rotational motion stop (instantaneously).

A professor doing a lecture demonstration stands at the center of a frictionless turntable, holding 5.00 -kg masses in each hand with arms extended so that each mass is \(1.20 \mathrm{~m}\) from his centerline. A (carefully selected!) student spins the professor up to a rotational speed of \(1.00 \mathrm{rpm} .\) If he then pulls his arms in by his sides so that each mass is \(0.300 \mathrm{~m}\) from his centerline, what is his new rotation rate? Assume that his rotational inertia without the masses is \(2.80 \mathrm{~kg} \mathrm{~m} / \mathrm{s}\), and neglect the effect on the rotational inertia of the position of his arms, since their mass is small compared to the mass of the body.

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?
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