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.)

Verified step by step guidance

1. Determine the angular velocity of the disk 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.
2. Calculate the moment of inertia of the disk. Use the formula for the moment of inertia of a disk rotating about an axis through its center, $I_{disk} = \frac{1}{2} M R^2$, where $M$ is the mass of the disk and $R$ is its radius.
3. Calculate the moment of inertia of the woman. Since the woman can be treated as a point mass at a distance equal to the radius of the disk from the axis, use the formula for the moment of inertia of a point mass, $I_{woman} = m r^2$, where $m$ is the mass of the woman and $r$ is the radius of the disk.
4. Add the moments of inertia of the disk and the woman to find the total moment of inertia of the system, $I_{total} = I_{disk} + I_{woman}$.
5. Calculate the total angular momentum of the system using the formula $L = I_{total} \omega$, where $\omega$ is the angular velocity calculated in step 1.

Key Concepts

Angular Momentum

Angular momentum is a measure of the rotational motion of an object and is defined as the product of an object's moment of inertia and its angular velocity. For a system of particles, the total angular momentum is the sum of the angular momentum of each particle. It is a vector quantity, which means it has both magnitude and direction, and is conserved in a closed system where no external torques are acting.

Moment of Inertia

The moment of inertia is a scalar value that quantifies how mass is distributed relative to an axis of rotation. It depends on both the mass of the object and the distance of the mass from the axis. For point masses, it is calculated as the product of the mass and the square of the distance from the axis. In this problem, the moment of inertia for the disk and the woman must be calculated separately and then combined to find the total moment of inertia of the system.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around an axis, expressed in radians per second or revolutions per second. It indicates the rate of change of angular position of an object. In this scenario, the disk's angular velocity is given as 0.80 revolutions per second, which can be converted to radians per second for calculations involving angular momentum.