An object made of two disk-shaped sections, \(\mathrm{A}\) and \(\mathrm{B}\), as shown in the figure, is rotating about an axis through the center of disk A. The masses and the radii of disks \(A\) and \(B\). respectively are, \(2.00 \mathrm{~kg}\) and \(0.200 \mathrm{~kg}\) and \(25.0 \mathrm{~cm}\) and \(2.50 \mathrm{~cm}\). a) Calculate the moment of inertia of the object. b) If the axial torque due to friction is \(0.200 \mathrm{~N} \mathrm{~m}\), how long will it take for the object to come to a stop if it is rotating with an initial angular velocity of \(-2 \pi \mathrm{rad} / \mathrm{s} ?\)

Angular velocity, denoted by the symbol \( \omega \), is a vector quantity that represents how fast an object rotates or revolves around a fixed axis. It is measured in radians per second (\( \text{rad/s} \)). In essence, it tells us the angle through which an object turns in a unit of time.

For instance, when a disk-shaped object like the one in the textbook exercise rotates, every point on the object moves through the same angle in the same time interval. This uniform motion describes the angular velocity of the object. When calculating the initial angular velocity \( \omega_0 = -2 \pi \ \text{rad/s} \) (negative indicating the direction of rotation), it gives us a complete understanding of how fast the object was spinning before any forces acted upon it.

When studying physics problems, knowing the angular velocity can help us understand the dynamic behavior of rotating systems and predict the outcomes of torque and angular acceleration applications.

Angular acceleration is the rate at which an object's angular velocity changes with time. Denoted by \( \alpha \), it is measured in radians per second squared (\( \text{rad/s}^2 \)). If an object's rotation speeds up or slows down, angular acceleration is present.

In our textbook problem, when an external torque due to friction is applied to the rotating object, it causes the object to experience angular acceleration opposite to its direction of rotation. The angular acceleration can be calculated by dividing the torque by the moment of inertia (\( \alpha = \frac{\tau}{I} \)), a key relationship that quantifies how the rotational motion is affected by external influences.

If an object moves with a constant angular velocity, there is no angular acceleration. However, when a force (i.e., a frictional torque) acts on the object, as in the case with our example, the decelerative angular acceleration affects how quickly the object comes to a stop. Understanding angular acceleration allows us to solve for time, given initial conditions and forces at play.

Torque, represented by the Greek letter \( \tau \), is a measure of the force that causes an object to rotate around an axis. It is the rotational equivalent of linear force and is calculated as the product of the force and the distance from the axis of rotation to the point where the force is applied, formatted as \( \tau = rF \sin(\theta) \), where \( r \) is the lever arm, \( F \) is the force, and \( \theta \) is the angle between the force and the lever arm direction.

In the given exercise, friction creates a torque that opposes the rotation of the object. The magnitude of this torque is \( 0.200 \ \text{N} \cdot \text{m} \) and acts to decelerate the object. Torque directly influences the angular acceleration of an object, since \( \tau = I \alpha \), where \( I \) is the moment of inertia. The larger the torque, the greater the object's change in rotational motion.

By analyzing torque, students can gain insights into the forces involved in rotational dynamics and the effects these forces have on an object's rotation, such as how long it takes for an object to come to a complete stop under a given torque.