It is the moment of friction force which brings the disc to...

It is the moment of friction force which brings the disc to rest. The force of friction is applied to each section of the disc, and since these sections lie at different distances from the axis, the moments of the forces of friction differ from section to section. To find $N_{z}$, where $z$ is the axis of rotation of the disc let us partition the disc into thin rings (Fig.). The force of friction acting on the considered element $d f r=k(2 \pi r d r \sigma) g$, (where $\sigma$ is the density of the disc) The moment of this force of friction is $d N_{z}=-r d f r=-2 \pi k \sigma g r^{2} d r$ Integrating with respect to $r$ from zero to $R$, we get $\boldsymbol{R}$ For the rotation of the disc about the stationary axis $z$, from the equation $N_{z}=I \beta_{z}$ $\pi k \sigma g R^{3}=\frac{\left(\pi R^{2} \sigma\right) R^{2}}{2} \beta_{z}$ or $\beta_{z}=-\frac{4 k g}{3 R}$ Thus from the angular kinematical equation $\omega_{z}=\omega_{0 z}+\beta_{z} t$ $0=\omega_{0}+\left(-\frac{4 k g}{3 R}\right) t$ or $t=\frac{3 R \omega_{0}}{4 k g}$

It is the moment of friction force which brings the disc to rest. The force of friction is applied to each section of the disc, and since these sections lie at different distances from the axis, the moments of the forces of friction differ from section to section. To find $N_{z}$, where $z$ is the axis of rotation of the disc let us partition the disc into thin rings (Fig.). The force of friction acting on the considered element $d f r=k(2 \pi r d r \sigma) g$, (where $\sigma$ is the density of the disc) The moment of this force of friction is $d N_{z}=-r d f r=-2 \pi k \sigma g r^{2} d r$
Integrating with respect to $r$ from zero to $R$, we get $\boldsymbol{R}$
For the rotation of the disc about the stationary axis $z$, from the equation $N_{z}=I \beta_{z}$ $\pi k \sigma g R^{3}=\frac{\left(\pi R^{2} \sigma\right) R^{2}}{2} \beta_{z}$ or $\beta_{z}=-\frac{4 k g}{3 R}$
Thus from the angular kinematical equation $\omega_{z}=\omega_{0 z}+\beta_{z} t$
$0=\omega_{0}+\left(-\frac{4 k g}{3 R}\right) t$ or $t=\frac{3 R \omega_{0}}{4 k g}$

Key Concepts

Integration in Rotational Dynamics

Integration is used in rotational dynamics to sum the contributions of infinitesimal elements of a continuous body. When forces or torques vary across an object (such as varying radii in a disc), integration allows one to account for these variations by summing the effects over the whole body. This method is essential for deriving expressions for total torque or moment of inertia from distributed parameters like density and geometry.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to angular acceleration about a given axis, analogous to mass in linear motion. It depends on the mass distribution relative to the axis—parts of the object far away from the rotation axis contribute more significantly. This concept is essential for linking the net applied torque to the angular acceleration of the object via the rotational form of Newton's second law.

Angular Kinematics

Angular kinematics deals with the relationships between angular displacement, angular velocity, and angular acceleration. Similar to linear kinematics, it describes the motion of rotating bodies through equations that relate these quantities over time. In problems involving frictional torque, angular kinematics is critical for determining how the angular velocity changes and eventually leads to the cessation of motion.

Torque

Torque, often known as the moment of a force, quantifies the tendency of a force to produce rotational motion about an axis. It depends on both the magnitude of the force and its distance from the pivot point. In problems involving distributed forces, such as friction acting over a body, torque is calculated for each element and summed (or integrated) to determine the overall effect on the system's rotation.

Friction Force

Friction is a resistive force that opposes motion between surfaces in contact. In the context of rotational motion, friction produces a torque that decelerates the rotating object. Understanding how friction acts on different parts of a rotating body is crucial, since its lever arm (distance from the axis) affects the generated torque, leading to non-uniform contributions across the body.

Transcript

Please give Ace some feedback