Question

A rotating disk has a mass of \(0.51 \mathrm{~kg}\), a radius of \(0.22 \mathrm{~m}\), and an angular speed of \(0.40 \mathrm{rad} / \mathrm{s}\). What is the angular momentum of the disk?

Short answer

The angular momentum of the disk is approximately \( 0.004937 \, \mathrm{kg} \cdot \mathrm{m}^2 / \, \mathrm{s} \).

Step-by-step solution

Identify the Formula for Angular Momentum

Angular momentum for a rotating disk (thin circular object) can be calculated using the formula \( L = I \omega \), where \( L \) is the angular momentum, \( I \) is the moment of inertia, and \( \omega \) is the angular speed in radians per second.

Calculate the Moment of Inertia for the Disk

For a disk, the moment of inertia \( I \) is given by \( I = \frac{1}{2} m r^2 \), where \( m \) is the mass, and \( r \) is the radius of the disk. Substitute \( m = 0.51 \, \mathrm{kg} \) and \( r = 0.22 \, \mathrm{m} \) into the equation:\[ I = \frac{1}{2} \times 0.51 \times (0.22)^2 \]

Compute the Moment of Inertia

Calculate the value of the moment of inertia using the values from the previous step:\[ I = \frac{1}{2} \times 0.51 \times 0.0484 = 0.012342 \, \mathrm{kg} \cdot \mathrm{m}^2 \]

Compute Angular Momentum

Substitute \( I = 0.012342 \, \mathrm{kg} \cdot \mathrm{m}^2 \) and angular speed \( \omega = 0.40 \, \mathrm{rad/s} \) into the formula \( L = I \omega \):\[ L = 0.012342 \times 0.40 = 0.0049368 \, \mathrm{kg} \cdot \mathrm{m}^2 / \, \mathrm{s} \]

Interpret the Result

The angular momentum of the disk is found to be \( 0.0049368 \, \mathrm{kg} \cdot \mathrm{m}^2 / \, \mathrm{s} \). This is a direct result of the moment of inertia and angular speed of the given disk.

Key concepts

Rotational Dynamics

Rotational dynamics deals with the study of the motion of objects that rotate around an axis. It's similar to linear dynamics but in circular motion. Understanding the concepts in rotational dynamics is critical for analyzing systems like wheels, gears, planets, and atoms.
At the heart of rotational dynamics, we find **angular momentum**, a quantity that depends on how much "rotational motion" an object has. The important thing is that just like linear momentum in linear motion, angular momentum is conserved in an isolated system. This means that if no external torque acts on an object, its angular momentum remains constant.

• Angular speed (\( \omega \)): This is how fast an object rotates, measured in radians per second.
• Angular momentum (\( L \)): This is the rotational equivalent of linear momentum and calculated like **\( L = I \omega \)**, where \( I \) is the moment of inertia.
• Torque: Analogous to force in linear motion, torque causes changes in an object's rotational motion.

Grasping these fundamentals can make solving rotational motion problems more intuitive. Remember, the principles of rotational dynamics are just extensions of linear dynamics, tailored for spinning objects.

Moment of Inertia

The moment of inertia is a crucial concept in understanding rotational motion. It tells us how much an object resists changes in its rotational motion - similar to how mass affects linear motion.
Mass distribution plays a huge role here. If more mass is located further from the rotational axis, the moment of inertia will be greater. This explains why figure skaters can spin much faster by pulling in their arms – they reduce their moment of inertia.

• Formula for Moment of Inertia (\( I \)): For a disk, \( I = \frac{1}{2} m r^2 \). This reflects the mass \( m \) and radius \( r \) impact on rotational resistance.
• Impacts on Rotational Motion: Higher moment of inertia means greater resistance to starting or stopping rotation.

By carefully calculating the moment of inertia, you can efficiently predict rotational behaviors and design systems accordingly. Remember, it's all about where the mass is in relation to the pivot point.

Circular Motion

Circular motion occurs when an object revolves around a fixed point or axis. This movement is crucial in many day-to-day and natural phenomena, from the spinning of a wheel to the orbit of planets.
There are several key elements essential for understanding circular motion:

• Centripetal force: Keeps the object moving in a circle. This force is directed towards the center of the circular path.
• Centrifugal force: Although not a real force, it's observed as the effect of inertia pushing an object outward as it moves in a circle.
• Angular velocity (\( \omega \)): This describes how quickly the angle changes as the object moves around the circle.

Objects under uniform circular motion experience a constant angular speed. However, any deviation (like speeding up or slowing down) involves angular acceleration. By mastering these elements, you gain insight into complex systems exhibiting circular motion in engineering, physics, and even biology. Remember, circular motion principles help explain and predict how and why objects move in a circular path.