(a) Let us indicate the forces and their points of...

(a) Let us indicate the forces and their points of application for the cylinder. Choosing the positive direction for $x$ and $\varphi$ as shown in the figure, we write the equation of motion of the cylinder axis and the equation of moments in the C.M. frame relative to that axis i.e. from equation $F_{x}=m w_{c}$ and $N_{z}=I_{c} \beta_{z}$ As there is no slipping of thread on the cylinder $w_{c}=\beta R$ From these three equations $T=\frac{m g}{6}=13 \mathrm{~N}, \beta=\frac{2}{5} \frac{g}{R}=5 \times 10^{2} \mathrm{rad} / \mathrm{s}^{2}$ (b) we have $\beta=\frac{2}{3} \frac{g}{R}$ So, $w_{c}=\frac{2}{3} g>0$ or, in vector form $\vec{w}_{c}=\frac{2}{3} \vec{g}$ $P=\vec{F} \cdot \vec{v}=\vec{F} \cdot\left(\vec{w}_{c} t\right)$ $=m \vec{g} \cdot\left(\frac{2}{3} \overrightarrow{g t}\right)=\frac{2}{3} m g^{2} t$

(a) Let us indicate the forces and their points of application for the cylinder. Choosing the positive direction for $x$ and $\varphi$ as shown in the figure, we write the equation of motion of the cylinder axis and the equation of moments in the C.M. frame relative to that axis
i.e. from equation $F_{x}=m w_{c}$ and $N_{z}=I_{c} \beta_{z}$
As there is no slipping of thread on the cylinder $w_{c}=\beta R$
From these three equations $T=\frac{m g}{6}=13 \mathrm{~N}, \beta=\frac{2}{5} \frac{g}{R}=5 \times 10^{2} \mathrm{rad} / \mathrm{s}^{2}$
(b) we have $\beta=\frac{2}{3} \frac{g}{R}$
So, $w_{c}=\frac{2}{3} g>0$ or, in vector form $\vec{w}_{c}=\frac{2}{3} \vec{g}$
$P=\vec{F} \cdot \vec{v}=\vec{F} \cdot\left(\vec{w}_{c} t\right)$
$=m \vec{g} \cdot\left(\frac{2}{3} \overrightarrow{g t}\right)=\frac{2}{3} m g^{2} t$

Key Concepts

Newton's Second Law

Newton's Second Law states that the net force acting on a body is equal to the mass of the body multiplied by its acceleration. This provides the fundamental relationship used to analyze the translational motion of any object under the influence of various forces.

Rotational Dynamics

Rotational dynamics applies Newton's laws to rotational motion, where the net torque (or moment) acting on an object is equal to the moment of inertia times its angular acceleration. This concept is essential for analyzing how forces cause objects to rotate about an axis.

Moment of Inertia

The moment of inertia is a property of an object that measures its resistance to changes in its rotational motion, analogous to mass in linear motion. It depends on how the mass is distributed relative to the axis of rotation and is key to determining the angular acceleration produced by a given torque.

Rolling Without Slipping

The rolling without slipping condition is a kinematic constraint that links the translational motion of an object to its rotational motion. Specifically, it relates the linear velocity (or acceleration) of the center of mass to the angular velocity (or angular acceleration) through the radius of the object, ensuring no relative motion at the point of contact.

Work and Power in Dynamics

Work and power in dynamics involve the conversion and rate of energy transfer within a system. Power is defined as the rate at which work is done and can be calculated as the dot product of the force vector and the velocity vector, providing a measure of the energy per unit time being used or transferred.

Please give Ace some feedback