How Does Gravity Affect the Fall of a Yo-Yo?

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The discussion centers on the dynamics of a yo-yo-like system where a disk falls under gravity while being influenced by a string. The key question is what conditions must be met for the disk's center of mass to fall vertically. Participants explore whether the disk would fall faster if it were attached to a pole. The conversation delves into the physics of rotational motion and gravitational effects on the system. Understanding these principles is crucial for analyzing the behavior of the yo-yo as it falls.
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Dear friends,

Here's a problem I can't solve. The disk on the picture falls under influence of gravity. The horizontal table creates no friction and the string that binds mass m and the disk is rolled up around the disk. The center of mass of the disk falls vertically.

My question is: what's the condition it has to satisfy, so that this last statement is true?

Do you think it's true that the disk will fall faster if it were attached to the pole?

Thanks for your help!
 

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