A yo-yo is made from two uniform disks, each with mass m and radius R, connected by a light axle of radius b. A light, thin string is wound several times around the axle and then held stationary while the yo-yo is released from rest, dropping as the string unwinds. Find the linear acceleration and angular acceleration of the yo-yo and the tension in the string.

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1 Concepts and Principles

1- $\textbf{Particle Under a Net Force}$ : If a particle of mass $\textcolor{black}{m}$ experiences a nonzero net force, its acceleration is related to the net force by $\textbf{Newton's second law}$ :

\begin{equation*} \sum F=ma \tag{1} \end{equation*}

2- $\textbf{Rigid Object Under a Net Torque}$ : If a rigid object free to rotate about a fixed axis has a net external torque acting on it, the object undergoes an angular acceleration $\alpha$ , where

\begin{gather*} \sum \tau_{\mathrm{ext}} = I \alpha \tag{2} \end{gather*}

This equation is the rotational analog to Newton's second law in the particle under a net force model.

3- The magnitude of the $\textbf{torque}$ associated with a force $\overrightarrow{\mathbf{F}}$ acting on an object at a distance $r$ from the rotation axis is:

\begin{gather*} \tau = r F \sin{\phi} = Fd \tag{3} \end{gather*}

where $\phi$ is the angle between the position vector of the point of application of the force and the force vector, and $d$ is the moment arm of the force, which is the perpendicular distance from the rotation axis to the line of action of the force.

4- When a rigid object rotates about a fixed axis, the angular acceleration are related to the translational acceleration through the relationship