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Why does a figure skater pull in her arms while increasing her angular velocity in a tight spin?

Short Answer

Expert verified
Answer: A figure skater's angular velocity increases when they pull their arms in during a tight spin due to the conservation of angular momentum. As the skater moves their arms closer to their body, their moment of inertia decreases. To maintain the conservation of angular momentum, their angular velocity must increase, resulting in a tighter and faster spin.

Step by step solution

01

Understand the conservation of angular momentum

Angular momentum, represented by the symbol L, is the rotational equivalent of linear momentum. It is defined as the product of an object's moment of inertia (I) and its angular velocity (w). When no external torques act on a system, the total angular momentum of the system remains constant. This is known as the conservation of angular momentum. Mathematically, we can express the conservation of angular momentum as: Initial angular momentum (Li) = Final angular momentum (Lf) Iiωi = Ifωf
02

Relate the skater's position to the moment of inertia

In this exercise, we need to understand how the skater's position affects her moment of inertia. The moment of inertia is a measure of the distribution of mass in an object relative to an axis of rotation. When a skater pulls in her arms, her mass is distributed closer to the axis of rotation, which decreases her moment of inertia.
03

Understand changes in the angular velocity

Since the conservation of angular momentum states that the initial and final angular momentum must be equal, as the skater moves her arms closer to her body, her moment of inertia (I) decreases. In order to maintain the conservation of angular momentum, her angular velocity (ω) must increase. Using the conservation of angular momentum equation from Step 1, we can write: Iiωi = Ifωf If If is less than Ii (skater pulls in arms), then ωf must be greater than ωi (angular velocity increases).
04

Explain the reason for increased angular velocity in a tight spin

By pulling in her arms, a figure skater reduces her moment of inertia, which in turn causes her angular velocity to increase. This allows the skater to spin tighter and faster. The increased angular velocity results in quicker, more impressive spins that are an essential part of figure skating performances. By understanding the conservation of angular momentum and the relationship between the moment of inertia and angular velocity, we can explain why a figure skater pulls in her arms while increasing her angular velocity in a tight spin.

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Key Concepts

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

Angular Momentum
Imagine spinning around with your arms stretched out and then pulling them in close to your body. You'll find yourself spinning faster, right? This is due to a principle called conservation of angular momentum. Angular momentum is a measure of how much rotation an object has. It's calculated by taking the object's moment of inertia (a way to measure how mass is spread out around an axis) and multiplying it by its angular velocity (how fast it's spinning).

For a spinning figure skater, when no outside forces are interfering, pulling in their arms doesn't change the total amount of angular momentum because it is conserved. So, when the moment of inertia decreases as the arms come in, the skater's angular velocity must increase to keep the angular momentum the same. This is why a skater spins faster with arms pulled in. To keep this simple, it essentially means if you're spinning and you want to go faster, bring your arms in, and if you want to slow down, stretch them out.
Moment of Inertia
The moment of inertia is a bit like the rotational version of mass. Think of it as 'rotational mass'. If you have a heavy object far from the center of rotation, it's harder to start or stop it spinning, giving it a high moment of inertia. Conversely, if the mass is close to the center, it's easier to spin, corresponding to a low moment of inertia.

For our figure skater, pulling her arms in shifts her body mass closer to the center of rotation (her spine), significantly reducing her moment of inertia. This change allows her to increase her rotational speed without having to put in extra effort. It's like tightening the nuts on a wheel – the closer everything is to the center, the easier it is to turn.
Angular Velocity
Now, let's talk about how fast things spin, which is known as angular velocity. You can think of angular velocity as the rate at which an object spins around a central point. Measured in radians per second, it's the rotational analog to linear velocity. In simpler terms, just as we say a car is moving at a certain speed, we can say a merry-go-round is spinning at a certain angular velocity.

When the figure skater brings her arms in, she's not only achieving grace and poise; she's demonstrating physics in action by increasing her angular velocity. This increase is essential for performing fast spins and is directly related to a decrease in her moment of inertia due to the conservation of angular momentum.
Rotational Motion
All of these concepts come together in rotational motion. Rotational motion is all about things that spin around a central axis – it could be a wheel, a planet, or a figure skater! This type of motion is governed by many of the same principles that apply to linear motion, but instead of moving in a straight line, objects are spinning.

Understanding rotational motion helps explain a variety of phenomena, from why the Earth keeps spinning to how a figure skater can dazzle with a fast pirouette. The intricacies of rotational motion are rich with physics and mathematics, giving us a framework to analyze and predict the movements of rotating objects.

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Most popular questions from this chapter

A disk of clay is rotating with angular velocity \(\omega .\) A blob of clay is stuck to the outer rim of the disk, and it has a mass \(\frac{1}{10}\) of that of the disk. If the blob detaches and flies off tangent to the outer rim of the disk, what is the angular velocity of the disk after the blob separates? a) \(\frac{5}{6} \omega\) b) \(\frac{10}{11} \omega\) c) \(\omega\) d) \(\frac{11}{10} \omega\) e) \(\frac{6}{5} \omega\)

Which one of the following statements concerning the moment of inertia of an extended rigid object is correct? a) The moment of inertia is independent of the axis of rotation. b) The moment of inertia depends on the axis of rotation. c) The moment of inertia depends only on the mass of the object. d) The moment of inertia depends only on the largest perpendicular dimension of the object.

The propeller of a light plane has a length of \(2.092 \mathrm{~m}\) and a mass of \(17.56 \mathrm{~kg}\). The rotational energy of the propeller is \(422.8 \mathrm{~kJ}\). What is the rotational frequency of the propeller (in rpm)? You can treat the propeller as a thin rod rotating about its center.

It is harder to move a door if you lean against it (along the plane of the door) toward the hinge than if you lean against the door perpendicular to its plane. Why is this so?

A uniform rod of mass \(M=250.0 \mathrm{~g}\) and length \(L=50.0 \mathrm{~cm}\) stands vertically on a horizontal table. It is released from rest to fall. a) What forces are acting on the rod? b) Calculate the angular speed of the rod, the vertical acceleration of the moving end of the rod, and the normal force exerted by the table on the rod as it makes an angle \(\theta=45.0^{\circ}\) with respect to the vertical. c) If the rod falls onto the table without slipping, find the linear acceleration of the end point of the rod when it hits the table and compare it with \(g\).

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