Problem 26 - College Physics

Question

\(\cdot\) A woman with mass 50.0 \(\mathrm{kg}\) is standing on the rim of a large disk that is rotating at 0.50 \(\mathrm{rev} / \mathrm{s}\) about an axis perpendicular to it through its center. The disk has a mass of 110 \(\mathrm{kg}\) and a radius of 4.0 \(\mathrm{m}\) . Calculate the magnitude of the total angular momentum of the woman-plus-disk system, assuming that you can treat the woman as a point.

Step-by-Step Solution

Answer

The total angular momentum is approximately 5280 kg·m²/s.

Step 1: Calculate the Woman's Moment of Inertia

The woman can be treated as a point mass. Therefore, her moment of inertia when standing on the rim is given by \( I_w = m_w \cdot r^2 \), where \( m_w = 50.0 \, \mathrm{kg} \) is her mass, and \( r = 4.0 \, \mathrm{m} \) is the radius of the disk. So, \( I_w = 50.0 \, \mathrm{kg} \times (4.0 \, \mathrm{m})^2 = 800 \, \mathrm{kg \cdot m^2} \).

Step 2: Calculate the Disk's Moment of Inertia

The disk's moment of inertia can be calculated using the formula for a solid disk, which is \( I_d = \frac{1}{2} m_d r^2 \), where \( m_d = 110 \, \mathrm{kg} \) is the mass of the disk. Hence, \( I_d = \frac{1}{2} \times 110 \, \mathrm{kg} \times (4.0 \, \mathrm{m})^2 = 880 \, \mathrm{kg \cdot m^2} \).

Step 3: Calculate the Total Moment of Inertia

The total moment of inertia of the woman-plus-disk system is the sum of their individual moments of inertia: \( I_{total} = I_w + I_d = 800 \, \mathrm{kg \cdot m^2} + 880 \, \mathrm{kg \cdot m^2} = 1680 \, \mathrm{kg \cdot m^2} \).

Step 4: Convert the Angular Velocity to Radians per Second

The angular velocity is given as \( 0.50 \, \mathrm{rev/s} \). Convert this to radians per second using the conversion factor \( 2\pi \, \mathrm{rad/rev} \). Therefore, \( \omega = 0.50 \, \mathrm{rev/s} \times 2\pi = \pi \, \mathrm{rad/s} \).

Step 5: Calculate the Total Angular Momentum

The total angular momentum of the system can now be calculated using \( L = I_{total} \times \omega \). Substituting the values, we get \( L = 1680 \, \mathrm{kg \cdot m^2} \times \pi \, \mathrm{rad/s} \approx 5280 \, \mathrm{kg \cdot m^2/s} \).

Key Concepts

Moment of Inertia

In physics, the moment of inertia is a measure of an object's resistance to rotational motion about an axis. The moment of inertia depends not only on the mass of the object but also on how that mass is distributed relative to the axis of rotation.
This concept is similar to the idea of mass in linear motion: while mass resists linear acceleration, the moment of inertia resists rotational acceleration.
For different shapes and mass distributions, the formulas for calculating moment of inertia vary. For instance:

• A point mass at a distance from the axis has a moment of inertia given by \( I = mr^2 \), where \( m \) is mass and \( r \) is the radius.
• A solid disk rotating about its center has a moment of inertia \( I = \frac{1}{2}mr^2 \), which considers the mass distributed evenly across the disk's surface.

In our example, both the woman (treated as a point mass) and the disk contribute to the total moment of inertia, calculated by summing their separate contributions. Understanding this principle is key to predicting how an object will behave under rotational forces.

Angular Velocity

Angular velocity is a measure of how fast an object rotates or revolves relative to another point, such as around an axis. Unlike linear velocity, which considers the motion in a straight line, angular velocity considers circular motion.
It is expressed in radians per second (rad/s), where one complete revolution around a circle corresponds to \( 2\pi \) radians.
To convert from revolutions per second to radians per second, it is essential to apply the conversion factor of \( 2\pi \). For example, if something is rotating at 0.50 revolutions per second, its angular velocity becomes \( 0.50 \times 2\pi = \pi \) rad/s.
Angular velocity is crucial to calculate other important rotational quantities, such as angular momentum. By knowing an object's angular velocity and moment of inertia, one can determine how much rotational movement an object possesses.

Solid Disk

A solid disk is a common geometrical object in rotational dynamics, often used as a model to understand rotational behavior in physics. A solid disk has its mass uniformly distributed across its volume, and this uniformity affects its moment of inertia.
For a solid disk, especially one rotating about an axis through its center and perpendicular to its face, the moment of inertia is calculated by \( I = \frac{1}{2}mr^2 \). This formula accounts for the disk's shape and consistent mass distribution across its radius.
Understanding the moment of inertia for a solid disk helps in many practical applications:

• Designing wheels and gears that need to rotate smoothly.
• Analyzing mechanical systems like flywheels and rotors used in engines and machinery.

In the example problem, both the woman's mass distribution and the disk's uniform mass play roles in computing the overall inertia and subsequently the system’s angular momentum. By assimilating these principles, one can better predict and model dynamic systems.