I was wondering how the angular and linear acceleration of a body in free fall interacted with each other. Let's imagine a man as a yoyo spinning around a string in free fall. It will have two accelerations, one angular and one linear. How is linear acceleration affected by angular acceleration? The linear acceleration is due $$\sum_{ext} F=T-mg.$$ Angular acceleration is due to the tension in the string and the friction between the string and the man: We have a torque due to gravity but I can not express mathematically what happens.
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$\begingroup$ Maybe this can help: youtube.com/watch?v=chC7xVDKl4Q $\endgroup$– Bob DCommented Aug 22, 2023 at 17:26
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$\begingroup$ A yo-yo is not a good example here because it has rather complicated dynamics when considering the general motion. $\endgroup$– jalexCommented Aug 22, 2023 at 19:22
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1 Answer
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The equations of motion on a plane still hold for a body like a yo-yo
$$ \begin{aligned} F_x & = m\, \ddot{x}_C \\ F_y & = m\, \ddot{y}_C \\ \tau_C & = I_C \alpha \end{aligned}$$
where $F_x$ is the net force acting on the body along the x-axis, $F_y$ the net force acting on the body along the y-axis, $\tau_C$ the net torque (including the effect of the forces) about the center of mass, $m$ the mass of the body, $\ddot{x}_C$ is the acceleration of the center of mass x-coordinate, $\ddot{y}_C$ is the acceleration of the center of mass y-coordinate, and $\alpha$ is the angular acceleration of the body.
