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Chapter 10: Problem 95

While the turntable is being accelerated, the person suddenly extends her legs. What happens to the turntable? (a) It suddenly speeds up; (b) it rotates with constant speed; (c) its acceleration decreases; (d) it suddenly stops rotating.

Short Answer

Expert verified
The correct answer is (c) its acceleration decreases.

Step by step solution

01

Understanding the Problem

First, you must understand that the turntable and the person are a closed system and the principle of conservation of angular momentum applies. This principle states that if no external torques act on a system, the total angular momentum of the system remains constant. In this case, the 'system' is the person and turntable combo.
02

Applying the Principle of Conservation of Angular Momentum

The angular momentum, which is the rotational counterpart of linear momentum, depends on two things: the rotational speed and the distribution of mass from the center of rotation--this is also known as the moment of inertia. If the person on the turntable extends her legs, she is changing the distribution of mass in the system, increasing the moment of inertia. According to the principle of conservation of angular momentum, if the moment of inertia increases and the angular momentum has to stay constant, then the angular speed must decrease.
03

Deducing the Outcome

Based on the decreased angular speed, we can answer the given question. The turntable shouldn't speed up, because that would mean an increase in angular speed. It wouldn't remain constant - the distribution of mass has changed and thus affected the speed of the turntable. The turntable wouldn't suddenly stop rotating, as this would not conserve angular momentum. Thus, the correct answer has to be (c) the acceleration (change in angular speed) of the turntable decreases.

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Key Concepts

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

Turntable Dynamics
In the realm of rotational motion, turntable dynamics is a fascinating topic. Imagine a turntable as a spinning platform. It can rotate when a force or torque acts on it. When discussing turntable dynamics, it's essential to note that it operates within a closed system.
A closed system means no external torques influence it. The turntable acts in conjunction with anyone or anything on it as a single system. This connects directly to the principle of conservation of angular momentum.
Angular momentum is the rotational equivalent of linear momentum. If the total angular momentum of the system stays the same, changes within the system adjust to maintain equilibrium.
  • An important concept is that as individuals on the turntable alter their position, they impact the system's dynamics.
  • For example, extending arms or legs changes the distribution of mass and affects how the turntable rotates.
  • Understanding these dynamics is essential for predicting how motion influences a turntable's behavior.
Moment of Inertia
The moment of inertia plays a significant role in rotational motion. It refers to how a mass's distribution impacts the resistance to change in rotational motion.
Think of it as the rotational equivalent of mass in linear motion. In a rotating system like a turntable, the moment of inertia determines how difficult it is to change the rotational speed.
When a person on a turntable extends their arms or legs, they increase the moment of inertia. The reason is that more mass moves farther from the rotation's center.
  • This change causes significant effects due to the conservation of angular momentum.
  • If the moment of inertia increases and angular momentum remains constant, something must change. That change is typically the rotational speed.
  • This concept is critical in engineering and physics, with applications extending to various fields.
Angular Speed
Angular speed is all about how fast an object rotates around a central point. It's measured in terms of how much angle (in radians) is covered per unit time.
In any rotating system, understanding angular speed is vital in predicting and explaining behavior changes.
When someone on a turntable extends their arms, the turntable's initial angular speed changes. This is due to the conservation of angular momentum. With an increase in the moment of inertia, the system compensates by reducing angular speed:
  • If angular speed decreases, the turntable rotates more slowly, even though the total angular momentum remains unchanged.
  • This is why activities like figure skating and gymnastics rely on pulling in or extending limbs to control rotational speed.
  • The change in angular speed showcases the connection and balance between moment of inertia and rotational speed in physics.

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Most popular questions from this chapter

A \(2.00 \mathrm{~kg}\) textbook rests on a frictionless, horizontal surface. A cord attached to the book passes over a pulley whose diameter is \(0.150 \mathrm{~m},\) to a hanging book with mass \(3.00 \mathrm{~kg} .\) The system is released from rest, and the books are observed to move \(1.20 \mathrm{~m}\) in \(0.800 \mathrm{~s}\) (a) What is the tension in each part of the cord? (b) What is the moment of inertia of the pulley about its rotation axis?

A hollow, thin-walled sphere of mass \(12.0 \mathrm{~kg}\) and diameter \(48.0 \mathrm{~cm}\) is rotating about an axle through its center. The angle (in radians) through which it turns as a function of time (in seconds) is given by \(\theta(t)=A t^{2}+B t^{4},\) where \(A\) has numerical value 1.50 and \(B\) has numerical value \(1.10 .\) (a) What are the units of the constants \(A\) and \(B ?\) (b) At the time \(3.00 \mathrm{~s}\), find (i) the angular momentum of the sphere and (ii) the net torque on the sphere.

A cord is wrapped around the rim of a solid uniform wheel \(0.250 \mathrm{~m}\) in radius and of mass \(9.20 \mathrm{~kg} .\) A steady horizontal pull of \(40.0 \mathrm{~N}\) to the right is exerted on the cord, pulling it off tangentially from the wheel. The wheel is mounted on frictionless bearings on a horizontal axle through its center. (a) Compute the angular acceleration of the wheel and the acceleration of the part of the cord that has already been pulled off the wheel. (b) Find the magnitude and direction of the force that the axle exerts on the wheel. (c) Which of the answers in parts (a) and (b) would change if the pull were upward instead of horizontal?

A thin uniform rod has a length of \(0.500 \mathrm{~m}\) and is rotating in a circle on a frictionless table. The axis of rotation is perpendicular to the length of the rod at one end and is stationary. The rod has an angular velocity of \(0.400 \mathrm{rad} / \mathrm{s}\) and a moment of inertia about the axis of \(3.00 \times 10^{-3} \mathrm{~kg} \cdot \mathrm{m}^{2}\). A bug initially standing on the rod at the axis of rotation decides to crawl out to the other end of the rod. When the bug has reached the end of the rod and sits there, its tangential speed is \(0.160 \mathrm{~m} / \mathrm{s}\). The bug can be treated as a point mass. What is the mass of (a) the rod; (b) the bug?

What fraction of the total kinetic energy is rotational for the following objects rolling without slipping on a horizontal surface? (a) A uniform solid cylinder; (b) a uniform sphere; (c) a thin-walled, hollow sphere; (d) a hollow cylinder with outer radius \(R\) and inner radius \(R / 2\).
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