(a) Let us choose the positive direction of the rotation angle φ, such that wc x and βz have ide

step 1 To solve this problem, I am drawing the diagram first. So the diagram looks something like this

(a) Let us choose the positive direction of the rotation...

(a) Let us choose the positive direction of the rotation angle $\varphi$, such that $w_{c x}$ and $\beta_{z}$ have identical signs (Fig.). Equation of motion, $F_{x}=m w_{c x}$ and $N_{c z}=I_{c} \beta_{z}$ gives : $F \cos \alpha-f r=m w_{c x}: f r R-F r=I_{c} \beta_{z}=\gamma m R^{2} \beta_{z}$ In the absence of the slipping of the spool $w_{c x}=\beta_{z} R$ From the three equations $w_{c x}=w_{c}=\frac{F[\cos \alpha-(r / R)]}{m(1+\gamma)}$ (b) As static friction $(f r)$ does not work on the spool, from the equation of the increment of mechanical energy $A_{\text {ex }}=\Delta T$. $A_{e x t}=\frac{1}{2} m v_{c}^{2}+\frac{1}{2} \gamma m R^{2} \frac{v_{c}^{2}}{R^{2}}=\frac{1}{2} m(1+\gamma) v_{c}^{2}$ $=\frac{1}{2} m(1+\gamma) 2 w_{c} x=\frac{1}{2} m(1+\gamma) 2 w_{c}\left(\frac{1}{2} w_{c} t^{2}\right)$ Note |that at $\cos \alpha=r / R$, there is no rolling and for $\cos \alpha<r / R, w_{c x}<0$, i.e. the spool will move towards negative $x$ -axis and rotate in anticlockwise sense.

(a) Let us choose the positive direction of the rotation angle $\varphi$, such that $w_{c x}$ and $\beta_{z}$ have identical signs (Fig.). Equation of motion, $F_{x}=m w_{c x}$ and $N_{c z}=I_{c} \beta_{z}$ gives :
$F \cos \alpha-f r=m w_{c x}: f r R-F r=I_{c} \beta_{z}=\gamma m R^{2} \beta_{z}$
In the absence of the slipping of the spool $w_{c x}=\beta_{z} R$ From the three equations $w_{c x}=w_{c}=\frac{F[\cos \alpha-(r / R)]}{m(1+\gamma)}$
(b) As static friction $(f r)$ does not work on the spool, from the equation of the increment of mechanical energy $A_{\text {ex }}=\Delta T$. $A_{e x t}=\frac{1}{2} m v_{c}^{2}+\frac{1}{2} \gamma m R^{2} \frac{v_{c}^{2}}{R^{2}}=\frac{1}{2} m(1+\gamma) v_{c}^{2}$
$=\frac{1}{2} m(1+\gamma) 2 w_{c} x=\frac{1}{2} m(1+\gamma) 2 w_{c}\left(\frac{1}{2} w_{c} t^{2}\right)$
Note |that at $\cos \alpha=r / R$, there is no rolling and for $\cos \alpha<r / R, w_{c x}<0$, i.e. the spool will move towards negative $x$ -axis and rotate in anticlockwise sense.

Key Concepts

Conservation of Mechanical Energy

In dynamics, conservation of mechanical energy involves accounting for both translational and rotational kinetic energies in a system free from non-conservative work. When static friction does no work, it allows the analysis of energy changes solely through changes in these kinetic energy forms. This principle simplifies the study of rolling objects by enabling energy-based methods to determine velocities and accelerations over time.

Torque and Moment of Inertia

Torque is the rotational equivalent of force and is responsible for angular acceleration about an axis. The moment of inertia quantifies an object's resistance to changes in its rotational state, analogous to mass in linear motion. Together, they form the fundamental relationship ? = I?, which is essential when analyzing the rotational behavior of bodies, especially in scenarios involving complex interactions between applied forces and geometry.

Static Friction and Its Role in Rolling Motion

Static friction is the force that prevents slipping between the rolling object and the surface. Although it does not perform work because there is no displacement at the point of contact, static friction is crucial for enforcing the rolling without slipping condition. It provides the necessary force (or torque) to align the rotational motion with the linear motion, ensuring a synchronized movement.

Rolling Without Slipping

This concept describes a situation where a rolling object moves such that the point of contact with the surface is momentarily at rest relative to the surface. It creates a fixed relation between the translational velocity (or acceleration) of the object’s center of mass and its angular velocity (or angular acceleration), ensuring that v = ?R. This condition is critical in analyzing the dynamics of rolling objects and connecting their rotational and translational motions.

Newton's Laws for Combined Translation and Rotation

Newton's laws are applied to both the translational motion (force equals mass times acceleration) and rotational motion (torque equals moment of inertia times angular acceleration) of an object. In systems involving rolling, these laws must be employed simultaneously to capture the complete dynamics, leading to coupled equations that relate linear forces, torques, and accelerations.

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