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Chapter 7: Problem 21

How does the law of conservation of angular momentum control a figure-skater's rate of spin?

Short Answer

Expert verified
A figure skater spins faster by pulling their arms in, decreasing their moment of inertia while conserving angular momentum.

Step by step solution

01

Understand the Law of Conservation of Angular Momentum

The law of conservation of angular momentum states that if no external torque acts on a system, the total angular momentum of that system remains constant. Angular momentum (L) can be calculated using the formula: \[ L = I \times \text{蠅} \] where \( I \) is the moment of inertia and \( \text{蠅} \) is the angular velocity.
02

Understand Moment of Inertia

Moment of inertia (\( I \)) depends on how the mass is distributed in relation to the axis of rotation. It's higher when the mass is farther from the axis and lower when the mass is closer. For a figure skater, extending arms increases \( I \) and pulling arms in decreases \( I \).
03

Relate Moment of Inertia to Angular Velocity

Since angular momentum \( L \) is conserved: \[ L = I_1 \text{蠅}_1 = I_2 \text{蠅}_2 \] where \( I_1 \) and \( \text{蠅}_1 \) are the moment of inertia and angular velocity initially, and \( I_2 \) and \( \text{蠅}_2 \) are after a change in position. If \( I \) decreases (by pulling arms in), \( \text{蠅} \) increases to keep \( L \) constant.
04

Apply to Figure Skating

When a figure skater starts a spin with arms extended, they have a higher moment of inertia and spin slowly (lower angular velocity). As they pull their arms in, the moment of inertia decreases, and they spin faster (higher angular velocity) to conserve angular momentum.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

angular momentum
Angular momentum is a key concept in the world of physics, especially when understanding rotating bodies like a figure skater. The angular momentum (
Moment of Inertia
The moment of inertia (
Angular velocity
Angular velocity (

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Most popular questions from this chapter

Using the exoplanet catalogs: a. Go to the "Catalog" Web page (http://exoplanet.eu/catalog) of the Extrasolar Planets Encyclopedia and set to "All Planets detected." Look for a star that has multiple planets. Make a graph showing the distances of the planets from that star, and note the masses and sizes of the planets. Put the Solar System planets on the same axis. How does this extrasolar planet system compare with the Solar System? b. Go to the "Exoplanets Data Explorer" website (http:// exoplanets.org and click on "Table." This website lists planets that have detailed orbital data published in scientific journals, and it may have a smaller total count than the website in part (a). Pick a planet that was discovered this year or last, as specified in the "First Reference" column. What is the planet's minimum mass? What is its semimajor axis and the period of its orbit? What is the eccentricity of its orbit? Click on the star name in the first column to get more information. Is there a radial velocity curve for this planet? Was it observed in transit, and if so, what is the planet's radius and density? Is it more like Jupiter or more like Earth?

Citizen science projects: a. Go to the "PlanetHunters" website at http://planethunters.org. PlanetHunters is part of the Zooniverse, a citizen science project that invites individuals to participate in a major science project using their own computers. To participate in this or any of the other Zooniverse projects mentioned in later chapters, you will need to sign up for an account. Read through the sections under 鈥淎bout," including the FAQ. What are some of the advantages to crowdsourcing Kepler data analysis? Back on the PlanetHunters home page, click on "Tutorial" and watch the "Introduction" and "Tutorial Video." When you're ready to try looking for planets, click on "Classify" and begin. Save a copy of your stars for your homework. b. Go to the "Disk Detective" website at http://www diskdetective.org/, another Zooniverse project for which you will need to make an account as in part (a). In this project, you will look at observations of young stars to see if there is evidence for a planetary disk. Under "Menu," read "Science" and 鈥淎bout," and then "Classify." Work through an example, and then classify a few images.

The best current technology can measure radial velocities of about \(0.3 \mathrm{m} / \mathrm{s}\). Suppose you are observing a spectral line with a wavelength of 575 nanometers (nm). How large a shift in wavelength would a radial velocity of \(0.3 \mathrm{m} / \mathrm{s}\) produce?

Jupiter has a mass equal to 318 times Earth's mass, an orbital radius of \(5.2 \mathrm{AU}\), and an orbital velocity of \(13.1 \mathrm{km} / \mathrm{s}\). Earth's orbital velocity is \(29.8 \mathrm{km} / \mathrm{s}\). What is the ratio of Jupiter's orbital angular momentum to that of Earth?

What is the source of the material that now makes up the Sun and the rest of the Solar System?
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