A woman with mass 50 kg is standing on the rim of a large disk that is rotating at 0.80 rev/s about an axis through its center. The disk has mass 110 kg and radius 4.0 m. Calculate the magnitude of the total angular momentum of the woman–disk system. (Assume that you can treat the woman as a point.)

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First, understand that the total angular momentum of the system is the sum of the angular momentum of the woman and the angular momentum of the disk. Angular momentum (L) is given by the formula: \( L = I \omega \), where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.

Calculate the angular velocity \( \omega \) in radians per second. Since the disk rotates at 0.80 revolutions per second, convert this to radians per second using the conversion factor \( 2\pi \) radians per revolution: \( \omega = 0.80 \times 2\pi \).

Determine the moment of inertia of the disk. For a solid disk rotating about its center, the moment of inertia \( I_{disk} \) is given by \( \frac{1}{2} m r^2 \), where \( m \) is the mass of the disk and \( r \) is its radius. Substitute the given values: \( m = 110 \text{ kg} \) and \( r = 4.0 \text{ m} \).

Calculate the moment of inertia of the woman, treated as a point mass at a distance \( r \) from the axis of rotation. The moment of inertia \( I_{woman} \) is given by \( m r^2 \), where \( m \) is the mass of the woman and \( r \) is the radius of the disk. Substitute the given values: \( m = 50 \text{ kg} \) and \( r = 4.0 \text{ m} \).

Add the angular momentum of the woman and the disk to find the total angular momentum of the system: \( L_{total} = I_{disk} \omega + I_{woman} \omega \). Use the previously calculated values for \( I_{disk} \), \( I_{woman} \), and \( \omega \) to find the total angular momentum.

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주요 개념

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

Angular Momentum

Angular momentum is a measure of the rotational motion of an object and is the product of its moment of inertia and angular velocity. For a system, it is the sum of the angular momentum of all components. It is conserved in the absence of external torques, making it crucial for analyzing rotational dynamics.

Moment of Inertia

Moment of inertia quantifies an object's resistance to changes in its rotational motion. It depends on the mass distribution relative to the axis of rotation. For a disk, it is calculated using the formula I = 0.5 * m * r^2, where m is mass and r is radius. This concept helps determine the angular momentum of rotating systems.

Rotational Kinematics

Rotational kinematics involves the study of motion parameters like angular velocity and angular displacement without considering forces. Angular velocity, measured in revolutions per second (rev/s), describes how fast an object rotates. Understanding these parameters is essential for calculating angular momentum in rotating systems.