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?

Ch. 11 - Angular Momentum; General Rotation

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

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Giancoli Douglas 5th edition

Ch. 11 - Angular Momentum; General Rotation

문제 14b

11장, 문제 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?

검증된 단계별 안내

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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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.