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Ch. 11 - Angular Momentum; General Rotation
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 11 - Angular Momentum; General RotationProblem 10b
Chapter 11, Problem 10b

A person of mass 75 kg stands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 920. kg·m². The platform rotates without friction with angular velocity 0.95 rad/s. The person walks radially to the edge of the platform. Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.

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Step 1: Understand the conservation of angular momentum. Since there is no external torque acting on the system, the total angular momentum of the system (platform + person) remains constant before and after the person walks to the edge. Angular momentum is given by \( L = I \omega \), where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.
Step 2: Calculate the initial moment of inertia of the system. The initial moment of inertia is the sum of the platform's moment of inertia and the person's contribution. The person is initially at the center, so their contribution to the moment of inertia is zero. Thus, \( I_{\text{initial}} = I_{\text{platform}} = 920 \ \text{kg·m}^2 \).
Step 3: Calculate the final moment of inertia of the system. When the person walks to the edge of the platform, their contribution to the moment of inertia is \( m r^2 \), where \( m \) is the person's mass and \( r \) is the radius of the platform. Thus, \( I_{\text{final}} = I_{\text{platform}} + m r^2 \). Substitute \( m = 75 \ \text{kg} \) and \( r = 3.0 \ \text{m} \) into the equation.
Step 4: Use the conservation of angular momentum to find the final angular velocity. Since \( L_{\text{initial}} = L_{\text{final}} \), we have \( I_{\text{initial}} \omega_{\text{initial}} = I_{\text{final}} \omega_{\text{final}} \). Solve for \( \omega_{\text{final}} \) using the known values of \( I_{\text{initial}} \), \( \omega_{\text{initial}} \), and \( I_{\text{final}} \).
Step 5: Calculate the rotational kinetic energy before and after the person’s walk. Rotational kinetic energy is given by \( KE = \frac{1}{2} I \omega^2 \). Compute \( KE_{\text{initial}} = \frac{1}{2} I_{\text{initial}} \omega_{\text{initial}}^2 \) and \( KE_{\text{final}} = \frac{1}{2} I_{\text{final}} \omega_{\text{final}}^2 \). Compare the two values to observe the change in rotational kinetic energy.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rotational Kinetic Energy

Rotational kinetic energy is the energy possessed by an object due to its rotation. It is calculated using the formula KE_rot = 0.5 * I * ω², where I is the moment of inertia and ω is the angular velocity. This concept is crucial for understanding how the energy of a rotating system changes when the distribution of mass changes, such as when a person moves on a merry-go-round.
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Moment of Inertia

Moment of inertia is a measure of an object's resistance to changes in its rotational motion, depending on the mass distribution relative to the axis of rotation. It is calculated by summing the products of each mass element and the square of its distance from the axis. In this scenario, the moment of inertia of the system will change as the person moves from the center to the edge of the platform, affecting the overall rotational kinetic energy.
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Conservation of Angular Momentum

Conservation of angular momentum states that if no external torque acts on a system, its total angular momentum remains constant. In the context of the merry-go-round, as the person walks outward, the moment of inertia increases, which causes the angular velocity to decrease to conserve angular momentum. This principle is essential for analyzing the changes in the system's rotational dynamics during the person's movement.
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Related Practice
Textbook Question

A woman of mass m stands at the edge of a solid cylindrical platform of mass M and radius R. At t = 0, the platform is rotating with negligible friction at angular velocity ω0 about a vertical axis through its center, and the woman begins walking with speed υ (relative to the platform) toward the center of the platform. Determine the angular velocity of the system as a function of time.

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Textbook Question

A woman of mass m stands at the edge of a solid cylindrical platform of mass M and radius R. At t = 0, the platform is rotating with negligible friction at angular velocity ω0 about a vertical axis through its center, and the woman begins walking with speed υ (relative to the platform) toward the center of the platform. What will be the angular velocity when the woman reaches the center?

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Textbook Question

A uniform disk turns at 4.1 rev/s around a frictionless central axis. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 11–32. They then turn together around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?

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Textbook Question

Show that  î x ĵ = k̂ , î x k̂ = - ĵ, and ĵ x k̂ = î.

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