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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: 9780137488179Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoGiancoli Douglas 5th editionCh. 11 - Angular Momentum; General RotationProblema 14b
Capitolo 11, Problema 14b

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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Start by identifying the principle of conservation of angular momentum. Since there is negligible friction, the total angular momentum of the system (woman + platform) is conserved.
Write the initial angular momentum of the system. The platform's moment of inertia is given by \( I_{platform} = \frac{1}{2} M R^2 \), and the woman's contribution to angular momentum is \( m R^2 \omega_0 \) (since she is initially at the edge). The total initial angular momentum is \( L_{initial} = \left( \frac{1}{2} M R^2 + m R^2 \right) \omega_0 \).
As the woman walks toward the center, her distance from the axis of rotation decreases, which changes her moment of inertia. At any point, her moment of inertia is \( I_{woman} = m r^2 \), where \( r \) is her distance from the center. The platform's moment of inertia remains constant.
Write the final angular momentum of the system when the woman reaches the center. At this point, her moment of inertia becomes zero (since \( r = 0 \)), and the platform's moment of inertia remains \( \frac{1}{2} M R^2 \). The final angular momentum is \( L_{final} = I_{platform} \omega_{final} = \frac{1}{2} M R^2 \omega_{final} \).
Set the initial angular momentum equal to the final angular momentum to solve for the final angular velocity \( \omega_{final} \). Use the equation \( \left( \frac{1}{2} M R^2 + m R^2 \right) \omega_0 = \frac{1}{2} M R^2 \omega_{final} \). Simplify and solve for \( \omega_{final} \).

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Conservation of Angular Momentum

The principle of conservation of angular momentum states that if no external torque acts on a system, the total angular momentum of that system remains constant. In this scenario, the initial angular momentum of the platform and the woman combined must equal the final angular momentum when the woman reaches the center, allowing us to relate the initial and final angular velocities.
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12:12
Conservation of Angular Momentum

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotation, depending on the mass distribution relative to the axis of rotation. For a system involving a rotating platform and a moving person, the moment of inertia will change as the woman moves toward the center, affecting the overall angular velocity of the system.
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Intro to Moment of Inertia

Relative Velocity

Relative velocity refers to the velocity of one object as observed from another object. In this problem, the woman's speed is given relative to the platform, which means we must account for both her movement and the platform's rotation to determine the final angular velocity when she reaches the center.
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04:27
Intro to Relative Motion (Relative Velocity)
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