Problem 65 - College Physics

Question

A 10 -kg rotating disk of radius \(0.25 \mathrm{~m}\) has an angular momentum of \(0.45 \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}\) What is the angular speed of the disk?

Step-by-Step Solution

Answer

The angular speed of the disk is 1.44 rad/s.

Step 1: Understand the relationship between angular momentum and angular speed

Angular momentum (L) of a rotating object is given by the formula \(L = I \times \omega\), where \(I\) is the moment of inertia, and \(\omega\) is the angular speed. Our goal is to find \(\omega\).

Step 2: Calculate the Moment of Inertia

For a solid disk, the moment of inertia \(I\) is given by \(I = \frac{1}{2} m r^2\). Plug in the values for mass \(m = 10 \, \text{kg}\) and radius \(r = 0.25 \, \text{m}\) into the equation to get \(I = \frac{1}{2} \times 10 \times (0.25)^2 = 0.3125 \, \text{kg} \cdot \text{m}^2\).

Step 3: Utilize the angular momentum formula to find angular speed

Given \(L = 0.45 \, \text{kg} \cdot \text{m}^2 / \text{s}\), use the relation \(L = I \times \omega\) to find \(\omega\). Rearranging gives \(\omega = \frac{L}{I}\). Substitute the calculated \(I\) and the given \(L\): \(\omega = \frac{0.45}{0.3125} = 1.44 \, \text{rad/s}\).

Key Concepts

Moment of Inertia

Moment of inertia is a fundamental property in rotational dynamics, which essentially measures an object's resistance to changes in its rotational state. You can think of it as the rotational equivalent of mass in linear motion. The moment of inertia depends on two main factors:

• Mass of the object, denoted as \(m\).
• The distribution of that mass relative to the axis of rotation.

For a solid disk, which is the shape in our exercise, the moment of inertia \(I\) can be calculated using the formula \(I = \frac{1}{2} m r^2\). Here, \(r\) represents the radius of the disk. By plugging in the mass and radius, you can easily find \(I\). This calculation shows how mass further from the center increases the moment of inertia, making it more challenging to change the disk's spinning behavior.
Understanding moment of inertia helps in predicting how different objects will behave under rotational motion, playing a critical role in engineering and physical simulations.

Angular Speed

Angular speed tells us how quickly an object rotates or spins around an axis. It's essentially the speed of rotation and is usually measured in radians per second (rad/s). This concept is closely linked to linear speed but for circular motion.

• Angular speed is denoted by the Greek letter \(\omega\).
• It can be thought of as how many radians the object covers in a unit of time.

Knowing the angular speed is crucial because it helps determine other dynamic properties of a rotating object, such as energy and momentum. In our example, once the moment of inertia \(I\) is known, we can easily find the angular speed by rearranging the formula for angular momentum, \(L = I \times \omega\), to solve for \(\omega = \frac{L}{I}\). This insight helps us comprehend rotational systems in fields like mechanics and electronics.

Rotating Disk

A rotating disk is a common physical model used to simplify and study rotational dynamics. By understanding its properties, we can learn how objects behave when they spin. In our example, a rotating disk is characterized by:

• Mass distribution, which affects the moment of inertia.
• Radius, which plays an integral role in calculating rotational dynamics.

When analyzing a rotating disk, we can use its simplicity to understand more complex systems that involve rotation, such as gears, wheels, or even celestial bodies like planets and stars. Through mathematical formulas, like those for moment of inertia and angular momentum, these discs serve as incredible learning tools to explain phenomena in physics and engineering.
Real-world applications include flywheels in vehicles or disks in data storage systems. These principles not only offer academic understanding but have practical significance in technology and industry.